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#### Question 1:

If PT is a tangent at T to a circle whose centre is O and OP = 17 cm, OT = 8 cm, Find the length of the tangent segment PT.

Let us put the given data in the form of a diagram.

We have to find TP. From the properties of tangents we know that a tangent will always be at right angles to the radius of the circle at the point of contact. Therefore is a right angle and triangle OTP is a right triangle.

We can find the length of TP using Pythagoras theorem. We have,

Therefore, the length of TP is 15 cm.

#### Question 2:

Find the length of a tangent drawn to a circle with radius 5 cm, from a point 13 cm from the centre of the circle.

Let us first put the given data in the form of a diagram.

We have to find TP. From the properties of tangents we know that a tangent will always be at right angles to the radius of the circle at the point of contact. Therefore is a right angle and triangle OTP is a right triangle.

We can find the length of TP using Pythagoras theorem. We have,

Therefore, the length of TP is 12 cm.

#### Question 3:

A point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 10 cm. Find the radius of the circle.

Let us put the given data in the form of a diagram.

We have to find OT. From the properties of tangents we know that a tangent will always be at right angles to the radius of the circle at the point of contact. Therefore is a right angle and triangle OTP is a right triangle.

We can find the length of TP using Pythagoras theorem. We have,

Therefore, the radius of the circle is 24 cm.

#### Question 4:

If from any point on the common chord of two interesting circle, tangents be drawn to the circles, prove that they are equal.

Let the two circles intersect at points X and Y. XY is the common chord.

Suppose A is a point on the common chord and AM and AN be the tangents drawn from A to the circle.

We need to show that AM = AN.

In order to prove the above relation, following property will be used.

“Let PT be a tangent to the circle from an external point P and a secant to the circle through P intersects the circle at points A and B, then PT = PA × PB”.

Now, AM is the tangent and AXY is a secant.

∴ AM2 = AX × AY ...(1)

AN is the tangent and AXY is a secant.

∴ AN2 = AX × AY ...(2)

From (1) and (2), we have

AM2 = AN2

∴ AM = AN

#### Question 5:

If the sides of a quadrilateral touch a circle. prove that the sum of a pair of opposite sides is equal to the sum of the other pair.

Let us first put the given data in the form of a diagram.

We have been asked to prove that the sum of the pair of opposite sides of the quadrilateral is equal to the sum of the other pair.

Therefore, we shall first consider,

AB + DC

But by looking at the figure we have,

AB + DC = AF + FB + DH + HC …… (1)

From the property of tangents we know that the length of two tangents drawn to a circle from a common external point will be equal. Therefore we have the following,

AF = AE

FB = BG

DH = ED

HC = CG

Replacing for all the above in equation (1), we have

AB + DC = AE + BG + ED + CG

AB + DC = (AE + ED) + (BG +CG)

AB + DC = AD + BC

Thus we have proved that the sum of the pair of opposite sides of the quadrilateral is equal to the sum of the other pair.

#### Question 6:

Out of the two concentric circles , the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle . Find the radius of the inner circle .

Let the centre of the two concentric circles be O.
CD is the tangent to the inner circle.
OP joins the centre of the circle to the tangent at the point of contant.
OP $\perp$ CD and OP bisects CD.
Thus, PD = 4 cm
In ∆OPD,

Hence, the radius of the inner circle = 3 cm.

#### Question 7:

A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle . Prove that R bisects the arc PRQ.

Given: Circle with centre O. PQ is the chord parallel to the tangent l at R

To prove: The point R bisects the arc PRQ.

Construction: Join OR intersecting PQ at S.

Proof:

OR ⊥ l (Radius is perpendicular to the tangent at the point of contact)

PQ || l     (given)

∴∠OSP = ∠OSQ = 90°  (pair of corresponding angles)

In ΔOPS and ΔOQS

OP = OQ  (Radii of the same circle)

OS = OS  (Common)

OSP = ∠OSQ  (Proved)

So,ΔOPS ≅ ΔOQS  (RHS congruence criterion)

⇒ ∠POS = ∠QOS  (C.P.C.T)

arc (PR) = arc (QR)  (Measure of the arc is same as the angle subtended by the arc at the centre)

Thus, the point R bisects the arc (PRQ).

#### Question 8:

Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at that point A .

Consider a circle with centre at O. l is tangent that touches the circle at painted to the circle at point A. Let AB is diameter. Consider a chord CD parallel to l. AB intersects CD at point Q.

To prove the given statement, it is required to prove CQ = QD
Since l is tangent to the circle at point A.
∴  AB ⊥ l
It is given that || CD
Since CD is a chord of the circle and OA ⊥ l.
Thus, OQ bisects the chord CD.
This means AB bisects chord CD.
This means CQ = QD.

#### Question 9:

If AB, AC, PQ are tangents in the given figure and AB = 5 cm, find the perimeter of Δ APQ.

We have been asked to find the perimeter of the triangle APQ.

Therefore,

Perimeter of ΔAPQ is equal to AP + AQ + PQ

By looking at the figure, we can rewrite the above as follows,

Let the Perimeter of ΔAPQ be P. So P= AP + AQ + PX + XQ

From the property of tangents we know that when two tangents are drawn to a circle from the same external point, the length of the two tangents will be equal. Therefore we have,

PX =PB

XQ =QC

Replacing these in the above equation we have,

P =AP + AQ + PB + QC

From the figure we can see that,

AP + PB = AB

AQ + QC = AC

Therefore, we have, P= AB + AC

It is given that AB = 5 cm.

Again from the same property of tangents we know that that when two tangents are drawn to a circle from the same external point, the length of the two tangents will be equal. Therefore we have,

AB = AC

Therefore,

AC = 5 cm

Hence,

P = 5 + 5=10

Thus the perimeter of triangle APQ is 10 cm.

#### Question 10:

Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

Given: XY and XY at are two parallel tangents to the circle with centre O and AB is the tangent at the point C, which intersects XY at A and XY at B.

To Prove: AOB = 90°

Construction: Let us joint point O to C.

Proof:

In ΔOPA and ΔOCA, we have

OP = OC (Radii of the same circle)

AP = AC (Tangents from point A)

AO = AO (Common side)

ΔOPA ΔOCA (SSS congruence criterion)

Therefore, POA = COA ……(i) (C.P.C.T)

Similarly, ΔOQB ΔOCB ……(ii)

Since POQ is a diameter of the circle, it is a straight line.

Therefore, POA + COA + COB + QOB = 180°

From equations (i) and (ii), it can be observed that

2COA + 2COB = 180°

∴ ∠COA + COB = 90°

So, AOB = 90°.

#### Question 11:

In the given figure, PQ is tangent at a point R of the circle with centre O. If ∠TRQ = 30°, find mPRS.

We have been given that.

From the property of tangents we know that a tangent will always be perpendicular to the radius at the point of contact. Therefore,

Looking at the given figure we can rewrite the above equation as follows,

We know that. Therefore,

Now consider. The two sides of this triangle OR and OT are nothing but the radii of the same circle. Therefore,

OR = OT

And hence is an isosceles triangle. We know that the angles opposite to the equal sides of the isosceles triangle will be equal. Therefore,

We have found out that. Therefore,

=

Now consider. We know that sum of all angles of a triangle will always be equal to . Therefore,

Now let us consider the straight line SOT. We know that the angle of a straight line is . Therefore,

From the figure we can see that,

That is,

=

We have found out that. Therefore,

Let us take up now. The sides SO and OR of this triangle are nothing but the radii of the same circle and hence they are equal. Therefore,is an isosceles triangle. In an isosceles triangle, the angles opposite to the two equal sides of the triangle will be equal. Therefore we have,

Also the sum of all angles of a triangle will be equal to. Therefore,

In the previous step we have found out that. Therefore,

Let us now take up . We know from the property of tangents that the angle between the radius of the circle and the tangent at the point of contact will be equal to . Therefore,

=

By looking at the figure we can rewrite the above equation as follows,

In the previous section we have found that. Therefore,

Thus we have found out that .

#### Question 12:

If PA and PB are tangents from an outside point P such that PA = 10 cm and ∠APB = 60°. Find the length of chord AB.

Let us first put the given data in the form of a diagram.

From the property of tangents we know that the length of two tangents drawn to a circle from a common external point will always be equal. Therefore,

PA=PB

Consider the triangle PAB. Since we have PA=PB, it is an isosceles triangle. We know that in an isosceles triangle, the angles opposite to the equal sides will be equal. Therefore we have,

Also, sum of all angles of a triangle will be equal to . Therefore,

Since we know that,

Now if we see the values of all the angles of the triangle, all the angles measure. Therefore triangle PAB is an equilateral triangle.

We know that in an equilateral triangle all the sides will be equal.

It is given in the problem that side PA = 10 cm. Therefore, all the sides will measure 10 cm. Hence, AB = 10 cm.

Thus the length of the chord AB is 10 cm.

#### Question 13:

In a right triangle ABC in which  $\angle$B= 900 , a circle is drawn with AB as diameter intersecting the hypotenuse AC at P . Prove that the tangent to the circle at P bisects BC.

Given: ΔABC is right triangle in which ∠ABC = 90°. A circle is drawn with side AB as diameter intersecting AC in P.

PQ is the tangent to the circle when intersects BC in Q.

Construction: Join BP.

Proof:

PQ and BQ are tangents drawn from an external point Q.

⇒ PQ = BQ                                      .....(i)   [Length of tangents drawn from an external point to the circle are equal]

⇒ ∠PBQ = ∠BPQ                                     [In a triangle, equal sides have equal angles opposite to them]

As , it is given that,

AB is the diameter of the circle.

∴ ∠APB = 90°                                              [Angle in a semi-circle is 90°]

APB + ∠BPC = 180°                                 [Linear pair]

⇒ ∠BPC = 180° – ∠APB = 180° – 90° = 90°

In ΔBPC,

BPC + ∠PBC + ∠PCB = 180°  [Angle sum property]

⇒ ∠PBC + ∠PCB = 180° – ∠BPC = 180° – 90° = 90°                ....(ii)

Now,

BPC = 90°

⇒ ∠BPQ + ∠CPQ = 90°                            .....(iii)

From (ii) and (iii), we get,

⇒ ∠PBC + ∠PCB = ∠BPQ + ∠CPQ

⇒ ∠PCQ = ∠CPQ                 [∠BPQ = ∠PBQ]

In ΔPQC,

PCQ = ∠CPQ

PQ = QC                            .....(iv)

From (i) and (iv), we get,

BQ = QC

Thus, tangent at P bisects the side BC.

#### Question 14:

From an external point P, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at a point E and PA = 14 cm, find the perimeter of Δ PCD.

Let us first put the given data in the form of a diagram.

It is given that PA = 14cm. we have to find the perimeter of.

Perimeter of is PC + CD + PD

Looking at the figure we can rewrite the equation as follows.

Perimeter of is PC + CE + ED + PD ……(1)

From the property of tangents we know that the length of two tangents drawn to a circle from the same external point will be equal. Therefore,

CE =CA

ED =DB

Replacing the above in equation (1), we have,

Perimeter of as PC + CA + DB + PD

By looking at the figure we get,

PC +CA =PA

DB +PD =PB

Therefore,

Perimeter of is PA + PB

It is given that PA = 14 cm. again from the same property of tangents which says that the length of two tangents drawn to a circle from the same external point will be equal, we have,

PA = PB

Therefore,

Perimeter of = 2PA

Perimeter of = 2 × 14

Perimeter of = 28

Thus perimeter of is 28 cm.

#### Question 15:

In the given figure, ABC is a right triangle right-angled at B such that BC = 6 cm and AB = 8 cm. Find the radius of its incircle.

From the property of tangents we know that the length of two tangents drawn to a circle from the same external point will be equal. Therefore, we have

BQ = BP

Let us denote BP and BQ by x

AP = AR

Let us denote AP and AR by y

RC = QC

Let us denote RC and RQ by z

We have been given that is a right triangle and BC = 6 cm and AB = 8 cm. let us find out AC using Pythagoras theorem. We have,

Consider the perimeter of the given triangle. We have,

AB + BC + AC = 8 + 6 + 10

AB + BC + AC = 24

Looking at the figure, we can rewrite it as,

AP + PB + BQ + QC + AR + RC = 24

Let us replace the sides with the respective x, y and z which we have decided to use.

Now, consider the side AC of the triangle.

AC = 10

Looking at the figure we can say,

AR + RC = 10

y + z = 10 …… (2)

Now let us subtract equation (2) from equation (1). We have,

x + y + z = 12

y + z = 10

After subtracting we get,

x = 2

That is,

BQ = 2, and

BP = 2

Now consider the quadrilateral BPOQ. We have,

BP = BQ (since length of two tangents drawn to a circle from the same external point are equal)

Also,

PO = OQ (radii of the same circle)

It is given that .

From the property of tangents, we know that the tangent will be at right angle to the radius of the circle at the point of contact. Therefore,

We know that sum of all angles of a quadrilateral will be equal to . Therefore,

Since all the angles of the quadrilateral are equal to and the adjacent sides also equal, this quadrilateral is a square. Therefore, all sides will be equal. We have found out that,

BP = 2 cm

PO = 2 cm

Thus the radius of the incircle of the triangle is 2 cm.

#### Question 16:

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

In the given figure, C is the mid point of the minor arc AB of the circle with centre O.
PQ is the tangent to the given circle through point C.
To Prove: Tangent drawn at the mid point of the arc $\stackrel{⏜}{AB}$ of a circle is parallel to the chord joining the end point of the arc $\stackrel{⏜}{AB}$.
i.e AB || PQ.
Proof: C is the mid point of the minor arc AB
⇒ minor arc AC = minor arc BC
⇒ AC = BC
Thus, ∆ABC is an iscosceles triangle.
Therefore, the perpendicular bisector of side AB of ∆ABC passes through the vertex C.
We know that the perpendicular bisector of a chord passes through the centre of the circle.
Since AB is a chord of the circle so, the perpendicular bisector of AB passes through the centre O.
Thus, it is clear that the perpendicular bisector of AB passes through the points O and C.
Therefore, $\mathrm{AB}\perp \mathrm{OC}$
Now, PQ is the tangent to the circle through the point C on the circle.
Therefore,

#### Question 17:

From a point P, two tangents PA and PB are drawn to a circle with centre O. If OP = diameter of the circle, show that Δ APB is equilateral.

Let us first put the given data in the form of a diagram.

Consider and. We have,

PO is the common side for both the triangles.

PA = PB(Tangents drawn from an external point will be equal in length)

OB = OA(Radii of the same circle)

Therefore, by SSS postulate of congruency, we have

ΔPOA ΔPOB

Hence,

…… (1)

Now let us consider and . We have,

PL is the common side for both the triangles.

(From equation (1))

PA = PB (Tangents drawn from an external point will be equal in length)

From SAS postulate of congruent triangles,

Therefore,

PL = LB …… (2)

Since AB is a straight line,

Let us now take up ΔOPB. We know that the radius of a circle will always be perpendicular to the tangent at the point of contact. Therefore,

By Pythagoras theorem we have,

It is given that

OP = diameter of the circle

Therefore,

OP = 2OB

Hence,

Consider . We have,

But we have found that,

Also from the figure, we can say

PL = PO − OL

Therefore,

…… (3)

Also, from , we have

…… (4)

Since Left Hand Sides of equation (3) and equation (4) are same, we can equate the Right Hand Sides of the two equations. Thus we have,

=

We know from the given data, that OP = 2.OB. Let us substitute 2OB in place of PO in the above equation. We get,

Substituting the value of OL and also PO in equation (3), we get,

Also from the figure we get,

AB = PL + LB

From equation (2), we know that PL = LB. Therefore,

AB = 2.LB

We have also found that. We know that tangents drawn from an external point will be equal in length. Therefore, we have

PA = PB

Hence,

PA =

Now, consider . We have,

PA =

PB =

AB =

Since all the sides of the triangle are of equal length, is an equilateral triangle. Thus we have proved.

#### Question 18:

Two tangent segments PA and PB are drawn to a circle with centre O such that ∠APB = 120°. Prove that OP = 2 AP.

Let us first put the given data in the form of a diagram. We have,

Consider and . We have,

Here, PO is the common side.

PA = PB (Length of two tangents drawn from the same external point will be equal)

OA = OB(Radii of the same circle)

By SSS congruency, we have is congruent to .

Therefore,

It is given that,

That is,

(Since )

In ,

(Since radius will be perpendicular to the tangent at the point of contact)

We know that,

Thus we have proved.

#### Question 19:

If Δ ABC is isosceles with AB = AC and C (O, r) is the incircle of the ΔABC touching BC at L,prove that L bisects BC.

Let us first put the given data in the form of a diagram.

It is given that triangle ABC is isosceles with

AB = AC …… (1)

By looking at the figure we can rewrite the above equation as,

AM + MB = AN + NC

From the property of tangents we know that the length of two tangents drawn to a circle from the same external point will be equal. Therefore,

AM = AN

Let us substitute AN with AM in the equation (1). We get,

AM + MB = AM + NC

MB = NC …… (2)

From the property of tangents we know that the length of two tangents drawn from the same external point will be equal. Therefore we have,

MB = BL

NC = LC

But from equation (2), we have found that

MB = NC

Therefore,

BL = LC

Thus we have proved that point L bisects side BC.

#### Question 20:

AB is a diameter  and AC is a chord of a circle with centre O such that  . The tangent at C intersects AB  at a point D . Prove that BC = BD.

It is given that ∠BAC = 30° and AB is diameter.

ACB = 90°       (Angle formed by the diameter is 90°)

In ∆ABC,

ACB + ∠BAC + ∠ABC = 180°

90° + 30° + ∠ABC = 180°

⇒ ∠ABC = 60°

⇒ ∠CBD = 180° – 60° = 120° ( ∠CBD and ∠ABC form a linear pair)

In ∆OCD,

OBC = ∠ABC = 60°

Since OB = OC, ∠OCB = ∠OBC = 60°      (OC = OB = radius)

In ∆OCB,

⇒ ∠COB + ∠OCB + ∠OBC = 180°

⇒ ∠COB + 60° + 60° = 180°

⇒ ∠COB = 60°

In ∆OCD,

COD + ∠OCD  + ∠ODC = 180°

60° + 90° + ∠ODC = 90°     (∠COD = ∠COB)

⇒ ∠ODC = 30°

In ∆CBD,

CBD = 120°

BDC = ∠ODC = 30°

⇒ ∠BCD + ∠BDC + ∠CBD = 180°

⇒ ∠BCD + 30° + 120° = 180°

⇒ ∠BCD + 30° = ∠BDC

Angles made by BC and BD on CD are equal, so ∆CBD is an isosceles triangle and therefore, BC = BD.

#### Question 21:

In the given figure, a circle touches all the four sides of a quadrilateral ABCD with AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.

The figure given in the question is below.

From the property of tangents we know that, the length of two tangents drawn from the same external point will be equal. Therefore we have the following,

SA = AP

For our convenience, let us represent SA and AP by a

PB = BQ

Let us represent PB and BQ by b

QC = CR

Let us represent QC and CR by c

DR = DS

Let us represent DR and DS by d

It is given in the problem that,

AB = 6

By looking at the figure we can rewrite the above equation as follows,

AP + PB = 6

a + b = 6

b = 6 − a …… (1)

Similarly we have,

BC = 7

BQ + QC = 7

b + c = 7

Let us substitute the value of b which we have found in equation (1). We have,

6 − a + c = 7

c = a + 1 …… (2)

CD = 4

CR + RD = 4

c + d = 4

Let us substitute the value of c which we have found in equation (2).

a + 1 + d = 4

a + d =3

As per our representations in the previous section, we can write the above equation as follows,

SA + DS = 3

By looking at the figure we have,

Thus we have found that length of side AD is 3 cm.

#### Question 22:

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

Let us first put the given data in the form of a diagram. Let us also draw another parallel tangent PQ parallel to the given tangent AB. Draw a radius OR to the point of contact of the tangent PQ. Let us also draw a radius CO parallel to the two tangents.

It is given that,

We know that sum of angles on the same side of the transversal will be equal to 180°. Therefore,

We know that the radius of the circle will always be perpendicular to the tangent at the point of contact. Therefore,

Again, since sum of angles on the same side of the transversal is equal to 180°, we have,

Now let us add equation (1) and (2). We get,

Looking at the figure we can rewrite the above equation as,

We know that the angle of a straight line will measure 180°. Therefore, ROS is the straight line. Also, RO is the radius which we have drawn and we know that a radius is always drawn from the centre of the circle. Therefore, line ROS passes through the centre of the circle.

Thus we have proved that the perpendicular to the tangent at the point of contact passes through the centre of the circle.

#### Question 23:

Two circles touch externally at a point P. From a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. Prove that TQ = TR.

We know that the lengths of tangents drawn from an external point to a circle are equal.

In the given figure, TQ and TP are tangents drawn to the same circle from an external point T.

∴ TQ = TP               .....(1)

Also, TP and TR are tangents drawn to the same circle from an external point T.

∴ TP = TR               .....(2)

From (1) and (2), we get

TQ = TR

#### Question 24:

A is a point at a distance 13 cm from the centre O of a circle  of radius  5 cm . AP and AQ  are the tangents to the circle at P and Q . If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect  AP at and AQ at C , find the perimeter of the $∆$ ABC .

A is a point 13 cm from centre O. AP and AQ are the tangents to the circle with centre O.
AP = AQ
In ∆APO,

Now In ∆ABC,
Perimeter = AB + BC + AC
= AB + BR + RC + AC
= AB + BP + CQ + AC           ( BR = BP, RC = CQ)
=  AP + AQ
= 12 + 12
= 24 cm

#### Question 25:

In the given figure, a circle is inscribed in a quadrilateral ABCD in which ∠B = 90°. It AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius r of the circle.

Let us first consider the quadrilateral OPBQ.

It is given that .

Also from the property of tangents we know that the radius of the circle will always be perpendicular to the tangent at the point of contact. Therefore we have,

We know that sum of all angles of a quadrilateral will always be equal to . Therefore,

OQ = OP (both are the radii of the same circle)

PB = BQ (from the property of tangents which says that the length of two tangents drawn to a circle from the same external point will be equal)

Since the adjacent sides of the quadrilateral are equal and also since all angles of the quadrilateral are equal to 90°, we can conclude that the quadrilateral OPBQ is a square.
It is given that DS = 5 cm.

From the property of tangents we know that the length of two tangents drawn from the same external point will be equal. Therefore,

DS = DR

DR = 5

It is given that,

DR + RA = 23

5 + RA = 23

RA = 18

Again from the same property of tangents we have,

RA = AQ

We have found out RA = 18. Therefore,

AQ = 18

It is given that AB = 29. That is,

AQ + QB = 29

18 + QB = 29

QB = 11

We have initially proved that OPBQ is a square. QB is one of the sides of the square. Since all sides of the square will be of equal length, we have,

OP = 11

OP is nothing but the radius of the circle.

Thus the radius of the circle is equal to 11 cm.

#### Question 26:

In the given figure, there are two concentric circles with centre O of radii 5 cm and 3 cm. From an external point P, tangent PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP.

The figure given the question is

From the property of tangents we know that the radius of the circle will always be perpendicular to the tangent at the point of contact. Therefore, OA is perpendicular to AP and triangle OAP is a right triangle. Therefore,

Now consider the smaller circle. Here again the radius OB will be perpendicular to the tangent BP. Therefore, triangle OBP is a right triangle. Hence we have,

Thus we have found that the length of BP is .

#### Question 27:

In the given figure, AB is a chord of length 16 cm of a circle of radius 10 cm. The tangents at A and B intersect at a point P. Find the length of PA.

Consider and .

From the property of tangents we know that the length of two tangents drawn form an external point will be equal. Therefore we have,

PA = PB

OB = OA (They are the radii of the same circle)

PO is the common side

Therefore, from SSS postulate of congruency, we have,

Hence,

…… (1)

Now consider and . We have,

(From (1))

PA is the common side.

From the property of tangents we know that the length of two tangents drawn form an external point will be equal. Therefore we have,

PA = PB

From SAS postulate of congruent triangles, we have,

Therefore,

LA = LB

It is given that AB = 16. That is,

LA + LB = 16

LA + LA = 16

2LA = 16

LA = 8

LB = 8

Also, ALB is a straight line. Therefore,

That is,

Since ,

Therefore,

Now let us consider. We have,

Consider. Here,

(Since the radius of the circle will always be perpendicular to the tangent at the point of contact)

Therefore,

…… (1)

Now consider

…… (2)

Since the Left Hand Side of equation (1) is same as the Left Hand Side of equation (2), we can equate the Right Hand Side of the two equations. Hence we have,

= …… (3)

From the figure we can see that,

OP = OL + LP

Therefore, let us replace OP with OL + LP in equation (3). We have,

We have found that OL = 6 and LB = 8. Also it is given that OB = 10. Substituting all these values in the above equation, we get,

Now, let us substitute the value of PL in equation (2). We get,

We know that tangents drawn from an external point will always be equal. Therefore,

PB = PA

Hence, we have,

PA =

#### Question 28:

In the given figure, PA and PB are tangents from an external point P to a circle with centre O. LN touches the circle at M. Prove that PL + LM = PN + MN.

The figure given in the question

From the property of tangents we know that the length of two tangents drawn from an external point will we be equal. Hence we have,

PA = PB

PL + LA = PN + NB …… (1)

Again from the same property of tangents we have,

LA = LM (where L is the common external point for tangents LA and LM)

NB = MN (where N is the common external point for tangents NB and MN)

Substituting LM and MN in place of LA and NB in equation (1), we have

PL + LM = PN + MN

Thus we have proved.

#### Question 29:

In the given figure, BDC is a tangent to the given circle at point D such that BD = 30 cm and CD = 7 cm. The other tangents BE and CF are drawn respectively from B and C to the circle and meet when produced at A making BAC a right angle triangle. Calculate (i) AF (ii) radius of the circle.

The given figure is below

(i) The given triangle ABC is a right triangle where side BC is the hypotenuse. Let us now apply Pythagoras theorem. We have,

Looking at the figure we can rewrite the above equation as follows.

…… (1)

From the property of tangents we know that the length of two tangents drawn from the same external point will be equal. Therefore we have the following,

BE = BD

It is given that BD = 30 cm. Therefore,

BE = 30 cm

Similarly,

CD = FC

It is given that CD = 7 cm. Therefore,

FC = 7 cm

Also, on the same lines,

EA = AF

Let us substitute these in equation (1). We get,

Therefore,

AF = 5

Or,

AF = − 42

Since length cannot have a negative value,

AF = 5

(ii) Let us join the point of contact E with the centre of the circle say O. Also, let us join the point of contact F with the centre of the circle O. Now we have a quadrilateral AEOF.

e

(Given in the problem)

(Since the radius will always be perpendicular to the tangent at the point of contact)

(Since the radius will always be perpendicular to the tangent at the point of contact)

We know that the sum of all angles of a quadrilateral will be equal to . Therefore,

Since all the angles of the quadrilateral are equal to 90° and the adjacent sides are equal, this quadrilateral is a square. Therefore all the sides are equal. We have found that

AF = 5

Therefore,

OD = 5

OD is nothing but the radius of the circle.

Thus we have found that AF = 5 cm and radius of the circle is 5 cm.

#### Question 30:

If  be the diameters of two concentric circle s and c be the length of a chord of a circle which is tangent to the other circle , prove that .

Let O be the centre of two concentric circles and PQ be the tangent to the inner circle that touches the circle at R.

Now, OQ= $\frac{1}{2}{d}_{2}$  and OR= $\frac{1}{2}{d}_{1}$

Also, PQ = c

As, PQ is the tangent to the circle.

⇒ OR ⊥ PQ

⇒ QR =$\frac{1}{2}\mathrm{PQ}=\frac{1}{2}c$

In Triangle OQR,

∴ By Pythagoras Theorem,

${\left(OQ\right)}^{2}={\left(OR\right)}^{2}+{\left(RQ\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒{\left(\frac{{d}_{2}}{2}\right)}^{2}={\left(\frac{{d}_{1}}{2}\right)}^{2}+{\left(\frac{c}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒{\left({d}_{2}\right)}^{2}={\left({d}_{1}\right)}^{2}+{c}^{2}$

#### Question 31:

In the given figure, tangents PQ and PR are drawn from an external point to a circle with centre O, such that $\angle \mathrm{RPQ}=30°$. A chord RS is drawn parallel to the tangent PQ. Find $\angle \mathrm{RQS}$.                                                                                                                                        [CBSE 2015]

It is given that tangents PQ and PR are drawn from an external point P to a circle with centre O, such that $\angle \mathrm{RPQ}=30°$. Also, RS || PQ.

Join OR and OS.

PQ and PR are tangents drawn from an external point P to a circle.

∴ PQ = PR          (Lengths of tangents drawn from an external point to a circle are equal)

In ∆PQR,

PQ = PR

$\angle \mathrm{PRQ}=\angle \mathrm{PQR}$     (In a triangle, equal sides have equal angles opposite to them)

Now,

$\angle \mathrm{PRQ}+\angle \mathrm{PQR}+\angle \mathrm{RPQ}=180°$      (Angle sum property)

Since SR || PQ and RQ is the transversal,

$\angle \mathrm{SRQ}=\angle \mathrm{RQP}=75°$      (Alternate angles)

Now, PR is the tangent and OR is the radius through the point of contact R.

$\angle \mathrm{ORP}=90°$       (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

In ∆SOR,

OR = OS     (Radii of the circle)

$\angle \mathrm{OSR}=\angle \mathrm{ORS}=60°$     (In a triangle, equal sides have equal angles opposite to them)

Now,

$\angle \mathrm{ORS}+\angle \mathrm{OSR}+\angle \mathrm{ROS}=180°$      (Angle sum property)
$⇒60°+60°+\angle \mathrm{ROS}=180°\phantom{\rule{0ex}{0ex}}⇒\angle \mathrm{ROS}=180°-120°=60°$

We know that the angle subtended by an arc at the centre is twice the angle subtended by it any point on the remaining part of the circle.

$\therefore \angle \mathrm{ROS}=2\angle \mathrm{RQS}\phantom{\rule{0ex}{0ex}}⇒\angle \mathrm{RQS}=\frac{1}{2}\angle \mathrm{ROS}=\frac{1}{2}×60°=30°$

#### Question 32:

From an external point P , tangents PA = PB are drawn to a circle with centre O   . If , then find $\angle AOB$.

It is given that PA and PB are tangents to the given circle.
$\therefore \angle \mathrm{PAO}=90°$      (Radius is perpendicular to the tangent at the point of contact.)
Now,
$\angle \mathrm{PAB}=50°$          (Given)
$\therefore \angle \mathrm{OAB}=\angle \mathrm{PAO}-\angle \mathrm{PAB}=90°-50°=40°$
In ∆OAB,
OB = OA    (Radii of the circle)
$\therefore \angle \mathrm{OAB}=\angle \mathrm{OBA}=40°$           (Angles opposite to equal sides are equal.)
Now,
$\angle \mathrm{AOB}+\angle \mathrm{OAB}+\angle \mathrm{OBA}=180°$       (Angle sum property)
$⇒\angle \mathrm{AOB}=180°-40°-40°=100°$

#### Question 33:

In the given figure, two tangents AB and AC are drawn to a circle with centre O such that ∠BAC = 120°. Prove that OA = 2AB.

Consider and.

We have,

OB = OC (Since they are radii of the same circle)

AB = AC (Since length of two tangents drawn from an external point will be equal)

OA is the common side.

Therefore by SSS congruency, we can say that andare congruent triangles.

Therefore,

It is given that,

We know that,

We know that,

Therefore,

=

OA = 2AB

#### Question 34:

The length of three concesutive sides of a quadrilateral circumscribing a circle are 4 cm, 5 cm, and 7 cm respectively. Determine the length of the fourth side.

Let us first put the given data in the form of a diagram.

From the property of tangents we know that the length of two tangents drawn from the same external point will be equal. Therefore we have,

AR = SA

Let us represent AR and SA by ‘a’.

Similarly,

QB = RB

Let us represent SD and DP by ‘b’

PC = CQ

Let us represent PC and PQ by ‘c’

SD = DP

Let us represent QB and RB by ‘d

It is given that,

AB = 4

AR + RB =4

a + b = 4

b = 4 − a …… (1)

Similarly,

BC = 5

That is,

b + c = 5

Let us substitute for b from equation (1). We get,

4 − a + c = 5

c − a = 1

c = a + 1 …… (2)

CD = 7

c + d = 7

Let us substitute for c from equation (2). We get,

a + 1 + d = 7

a + d = 6

In the previous section we had represented AS and SR with ‘a’ and SD and DP with ‘b’. We shall now put AS in place of ‘a’ and SD in place of ‘d’. We get,

AS + SD = 6

Therefore, the length of the fourth side of the quadrilateral is 6 cm.

#### Question 35:

The common tangents AB and CD to two circles with centres O and O' intersect at E between their centres . Prove that the  points E and O' are collinear .

Here Angle AEC and DEB are equal ( vertically opposite angles)

Join OA and OC,

So in triangle OAE and OCE, we have

OA = OC ( radii of same circle)

OE = OE  (common)

$\angle$OAE = $\angle$OCE   [90 each, as tangent is always perpendicular to its radius at point of contact]

So, all four angles are equal and bisected by OE and OE'.
Hence, O, E' and O' are collinear.

#### Question 36:

In the given figure, common tangents PQ and RS to two circles intersect at A. Prove that PQ = RS.

The figure given in the question is

We know from the property of tangents that the length of two tangents drawn from a common external point will be equal. Therefore,

PA = RA …… (1)

AQ = AS …… (2)

Let us add equation (1) and (2)

PA + AQ = RA + AS

PQ = RS

Thus we have proved that PQ = RS.

#### Question 37:

Two concentric circles are of diameters 30 cm and 18 cm. Find the length of the chord of the larger circle which touches the smaller circle.                                                                                                                                                                                       [CBSE 2014]

Let O be the common centre of the two circles and AB be the chord of the larger circle which touches the smaller circle at C.

Join OA and OC. Then,

OC = $\frac{18}{2}$ cm = 9 cm and OA = $\frac{30}{2}$ cm = 15 cm

We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Also, the perpendicular drawn from the centre of a circle to a chord bisects the chord.

∴ OC ⊥ AB and C is the mid-point of AB.

In right ∆OCA,

∴ AB = 2AC = 2 × 12 cm = 24 cm

Thus, the required length of the chord is 24 cm.

#### Question 38:

AB and CD are common tangents to two circles of equal radii. Prove that AB = CD.

Given: Two circles with centre’s O and O'. AB and CD are common tangents to the circles which intersect in P.

To Prove: AB = CD

Proof:

AP = PC (length of tangents drawn from an external point to the circle are equal)                              ..… (1)

PB = PD (length of tangents drawn from an external point to the circle are equal)                              ..… (2)

Adding (1) and (2), we get

AP + PB = PC + PD

AB = CD
Hence Proved

#### Question 39:

A triangle PQR is drawn to circumscribe a circle of radius 8 cm such that the segments QT and TR, into which QR is divided by the point of contact T, are of lengths 14 cm and 16 cm respectively. If area of ∆PQR is 336 cm2, find the sides PQ and PR.                           [CBSE 2014]

Here, T, S and U are the points of contact of the circle with the sides QR, PQ and PR, respectively.

OT = OS = OU = 8 cm      (Radii of the circle)

We know that the lengths of tangents drawn from an external point to a circle are equal.

∴ QS = QT = 14 cm

RU = RT = 16 cm

PS = PU = x cm (say)

So, QR = QT + TR = 14 cm + 16 cm = 30 cm

PQ = PS + SQ = x cm + 14 cm = (x + 14) cm

PR = PU + UR = x cm + 16 cm = (x + 16) cm

Also, the tangent at any point of a circle is perpendicular to the radius through the point of contact.

∴ OT ⊥ QR, OS ⊥ PQ and OU ⊥ PR

Now,

ar(∆OQR) + ar(∆OPQ) + ar(∆OPR) = ar(∆PQR)

$⇒8x=336-240=96\phantom{\rule{0ex}{0ex}}⇒x=12$

∴ PQ = (x + 14) cm = (12 + 14) cm = 26 cm

PR = (x + 16) cm = (12 + 16) cm = 28 cm

Hence, the lengths of sides PQ and PR are 26 cm and 28 cm, respectively.

#### Question 40:

In Fig . 10.69, the tangent at a point C of a circle and a diameter AB when extended intersect at P . If $\angle$PCA =1100, find  $\angle$CBA. [Hint: Join CO.]
figure

Since, OC = OA           (radii of the circle)

#### Question 41:

AB is a chord of a circle with centre O , AOC  is a diameter and AT is the tangent at A as shown in Fig . 10.70. Prove that $\angle$BAT$\angle$ACB

In the given figure,
AC is the diameter.
So,              (Angle formed by the diameter on the circle is 90º)
AT is the tangent at point A.
Thus, $\angle \mathrm{CAT}=90°$
In ∆ABC,

Hence Proved

#### Question 42:

In the given figure, a ∆ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 8 cm and 6 cm respectively. Find the lengths of sides AB and AC, when area of ∆ABC is 84 cm2.                                                                 [CBSE 2015]

Here, D, E and F are the points of contact of the circle with the sides BC, AB and AC, respectively.

OD = OE = OF = 4 cm      (Radii of the circle)

We know that the lengths of tangents drawn from an external point to a circle are equal.

∴ BD = BE = 8 cm

CD = CF = 6 cm

AE = AF = x cm (say)

So, BC = BD + CD = 8 cm + 6 cm = 14 cm

AB = AE + BE = x cm + 8 cm = (x + 8) cm

AC = AF + FC = x cm + 6 cm = (x + 6) cm

Also, the tangent at any point of a circle is perpendicular to the radius through the point of contact.

∴ OD ⊥ BC, OE ⊥ AB and OF ⊥ AC

Now,

ar(∆OBC) + ar(∆OAB) + ar(∆OCA) = ar(∆ABC)

$⇒4x=84-56=28\phantom{\rule{0ex}{0ex}}⇒x=7$

∴ AB = (x + 8) cm = (7 + 8) cm = 15 cm

AC = (x + 6) cm = (7 + 6) cm = 13 cm

Hence, the lengths of sides AB and AC are 15 cm and 13 cm, respectively.

#### Question 43:

In the given figure, AB is a diameter of a circle with centre O and AT is a tangent. If $\angle$AOQ = 58º, find $\angle$ATQ.                        [CBSE 2015]

It is given that $\angle$AOQ = 58º.

We know that the angle subtended by an arc at the centre is twice the angle subtended by it any point on the remaining part of the circle.

$\angle \mathrm{ABQ}=\frac{1}{2}\angle \mathrm{AOQ}=\frac{1}{2}×58°=29°$

Now, AT is the tangent and OA is the radius of the circle through the point of contact A.

$\therefore \angle \mathrm{OAT}=90°$    (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

In ∆ABT,

(Angle sum property)

#### Question 44:

In the given figure, OQ : PQ = 3.4 and perimeter of Δ POQ = 60 cm. Determine PQ, QR and OP.

In the figure,

. Therefore we can use Pythagoras theorem to find the side PO.

…… (1)

In the problem it is given that,

…… (2)

Substituting this in equation (1), we have,

…… (3)

It is given that the perimeter of is 60 cm. Therefore,

PQ + OQ + PO = 60

Substituting (2) and (3) in the above equation, we have,

Substituting for PQ in equation (2), we have,

OQ is the radius of the circle and QR is the diameter. Therefore,

QR = 2OQ

QR = 30

Substituting for PQ in equation (3), we have,

Thus we have found that PQ = 20 cm, QR = 30 cm and PO = 25 cm.

#### Question 45:

Equal circles with centres O and O' touch each other at X. OO' produced to meet a circle with centre O', at A. AC is a tangent to the circle whose centre is O. O'D is perpendicular to AC. Find the value of $\frac{DO\text{'}}{CO}$.

Consider the two triangles and .

We have,

is a common angle for both the triangles.

(Given in the problem)

(Since OC is the radius and AC is the tangent to that circle at C and we know that the radius is always perpendicular to the tangent at the point of contact)

Therefore,

From AA similarity postulate we can say that,

~

Since the triangles are similar, all sides of one triangle will be in same proportion to the corresponding sides of the other triangle.

Consider AO of and AO of .

Since AO and OX are the radii of the same circle, we have,

AO = OX

Also, since the two circles are equal, the radii of the two circles will be equal. Therefore,

AO = XO

Therefore we have

Since ~,

We have found that,

Therefore,

#### Question 46:

In the given figure, BC is a tangent to the circle with centre O. OE bisects AP. Prove that ΔAEO $~$Δ ABC.

The figure given in the question is below

Let us first take up .

We have,

OA = OP (Since they are the radii of the same circle)

Therefore, is an isosceles triangle. From the property of isosceles triangle, we know that, when a median drawn to the unequal side of the triangle will be perpendicular to the unequal side. Therefore,

Now let us take up and .

We know that the radius of the circle will always be perpendicular to the tangent at the point of contact. In this problem, OB is the radius and BC is the tangent and B is the point of contact. Therefore,

Also, from the property of isosceles triangle we have found that

Therefore,

=

is the common angle to both the triangles.

Therefore, from AA postulate of similar triangles,

~

Thus we have proved.

#### Question 47:

In the given figure, PO $\perp$ QO. The tangents to the circle at P and Q intersect at a point T. Prove that PQ and OT are right bisector of each other.

In the given figure,

PO = OQ (Since they are the radii of the same circle)

PT = TQ (Length of the tangents from an external point to the circle will be equal) Now considering the angles of the quadrilateral PTQO, we have,

(Given in the problem)

(The radius of the circle will be perpendicular to the tangent at the point of contact)

(The radius of the circle will be perpendicular to the tangent at the point of contact)

We know that the sum of all angles of a quadrilateral will be equal to . Therefore,

Thus we have found that all angles of the quadrilateral are equal to 90°.

Since all angles of the quadrilateral PTQO are equal to 90° and the adjacent sides are equal, this quadrilateral is a square.

We know that in a square, the diagonals will bisect each other at right angles.

Therefore, PQ and OT bisect each other at right angles.

Thus we have proved.

#### Question 48:

In the given figure, O is the centre of the circle and BCD is tangent to it at C. Prove that ∠BAC + ∠ACD = 90°.

In the given figure, let us join D an A.

Consider . We have,

OC = OA (Radii of the same circle)

We know that angles opposite to equal sides of a triangle will be equal. Therefore,

…… (1)

It is clear from the figure that

Now from (1)

Now as BD is tangent therefore

Therefore

From the figure we can see that

Thus we have proved.

#### Question 49:

Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines .

Let  be two intersecting lines. Suppose a circle with centre O touches the lines  at M and N respectively.

OM = ON    (Radius of the same circle)

O is equidistant form .

In ΔOPM and ΔOPN,

OMP =  ∠ONP  (Radius is perpendicular to the tangent at the point of contact)

OP = OP  (Common)

OM = ON    (Radius of the same circle)

∴ ΔOPM  ΔOPN    (RHS congruence criterion)

⇒ ∠MPO = ∠NPO  (CPCT)

l bisects ∠MPN.

O lies on the bisector of the angles between  i.e., O lies on l.

Thus, the centre of the circle touching two intersecting lines lies on the angle bisector of the two lines.

#### Question 50:

In Fig. 8.78, there are two concentric circles with centre O. PRT and PQS are tangents to the inner circle from a point P lying on the outer circle. If PR = 5 cm, find the length of PS.

Given that PR = 5 cm.
PR and PQ are the tangents to the inner circle so,
PR = PQ = 5 cm                                (Tangents drawn from an external point to the circle are equal)
Now draw a perpendicular from the centre O to the tangent PS.
PS is the chord of the inner circle. we know that the perpendicular drawn
from the centre of the circle to the chord bisects the chord. So, PQ = QS = 5 cm
PS = PQ + QS = 5 cm + 5 cm = 10 cm

#### Question 51:

In Fig. 8.79, PQ is a tangent from an external point P to a circle with centre O and OP cuts the circle at T and QOR is a diameter. If ∠POR = 130° and S is a point on the circle, find ∠1 + ∠2.

Given: ∠POR = 130°
So, ∠TSR = $\frac{1}{2}\angle \mathrm{POR}=\frac{1}{2}×130°=65°=\angle 2$      .....(1)                   (Since angle subtended by the arc at the centre is double   the angle subtended by it at the remaining part of the circle)
∠POQ = 180º − ∠POR = 180º − 130º = 50º      .....(2)          (Linear pair)
In $△$POQ,
$\angle 1+\angle \mathrm{POQ}+\angle \mathrm{OQP}=180°\phantom{\rule{0ex}{0ex}}⇒\angle 1+50°+90°=180°\phantom{\rule{0ex}{0ex}}⇒\angle 1=40°$

#### Question 52:

In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If PA = 12 cm, QC = QD = 3 cm, then find PC + PD.

Given: PA and PB are the tangents to the circle.
PA = 12 cm
QC = QD = 3 cm

To find: PC + PD

PA = PB = 12 cm                (The lengths of tangents drawn from an external point to a circle are equal)
Similarly, QC = AC = 3 cm
and QD = BD = 3 cm.

Now, PC = PA − AC = 12 − 3 = 9 cm
Similarly, PD = PB − BD = 12 − 3 = 9 cm

Hence, PC + PD = 9 + 9 = 18 cm.

#### Question 1:

Mark the correct alternative in each of the following:

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q such that OQ = 12 cm. Length PQ is

(a) 12 cm

(b) 13 cm

(c) 8.5 cm

(d)

Let us first put the given data in the form of a diagram.

Given data is as follows:

OQ = 12 cm

OP = 5 cm

We have to find the length of QP.

We know that the radius of a circle will always be perpendicular to the tangent at the point of contact. Therefore, OP is perpendicular to QP. We can now use Pythagoras theorem to find the length of QP.

Therefore the correct answer is choice (d).

#### Question 2:

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is

(a) 7 cm

(b) 12 cm

(c) 15 cm

(d) 24.5 cm

Let us first put the given data in the form of a diagram.

The given data is as follows:

QP = 24 cm

QO = 25 cm

We have to find the length of OP, which is the radius of the circle.

We know that the radius of a circle will always be perpendicular to the tangent at the point of contact. Therefore, OP is perpendicular to QP. We can now use Pythagoras theorem to find the length of QP.

Therefore the length of the radius of the circle is 7 cm.

Hence the correct answer to the question is choice (a).

#### Question 3:

The length of the tangent from a point A at a circle, of radius 3 cm, is 4 cm. The distance of A from the centre of the circle is

(a)

(b) 7 cm

(c) 5 cm

(d) 25 cm

Let us first put the given data in the form of a diagram.

Given data is as follows:

OP = 3 cm

AP = 4 cm

We have to find the length of OA.

We know that the radius of a circle will always be perpendicular to the tangent at the point of contact. Therefore, OP is perpendicular to AP. We can now use Pythagoras theorem to find the length of OA.

Therefore, the distance of the point A from the center of the circle is 5 cm.

Hence the correct answer is choice (c).

#### Question 4:

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at

(a) 50°

(b) 60°

(c) 70°

(d) 80°

Let us first put the given data in the form of a diagram.

Given data is as follows:

We have to find.

We know that the radius of the circle will always be perpendicular to the tangent at the point of contact. Therefore,

Also, we know that sum of all angles of a quadrilateral will always be equal to 360°. Therefore in quadrilateral AOBP, we have,

Now consider ΔPOA and ΔPOB. We have,

PO is the common side for both the triangles.

PA = PB (Length of tangents drawn from the same external point will be equal)

OA = OB(Radii of the same circle)

Therefore, from SSS postulate of congruent triangles,

Therefore,

We have found that

That is,

Therefore, the correct answer is choice (a).

#### Question 5:

If TP and TQ are two tangents to a circle with centre O so that ∠POQ = 110°, then, ∠PTQ is equal to

(a) 60°

(b) 70°

(c) 80°

(d) 90°

Let us first put the given data in the form of a diagram.

It is given that,

We have to find

We know that the radius of the circle will always be perpendicular to the tangent at the point of contact. Therefore,

Also, we know that sum of all angles of a quadrilateral will always be equal to 360°. Therefore in quadrilateral TQOP, we have,

Therefore, the correct answer to this question is choice (b).

#### Question 6:

PQ is a tangent to a circle with centre O at the point P. If Δ OPQ is an isosceles triangle, then ∠OQP is equal to

(a)  30°

(b) 45°

(c) 60°

(d) 90°

Let us first put the given data in the form of a diagram.

We know that the radius of a circle will always be perpendicular to the tangent at the point of contact. Therefore, OP is perpendicular to QP. Therefore,

The side opposite tois OQ.OQ will be the longest side of the triangle. So, in the isosceles right triangle ΔOPQ,

OP = PQ

And the angles opposite to these two sides will also be equal. Therefore,

We know that sum of all angles of a triangle will always be equal to 180°. Therefore,

Therefore, the correct answer to this question is choice (b).

#### Question 7:

Two equal circles touch each other externally at C and AB is a common tangent to the circles . Then, ∠ACB =

(a) 60°

(b) 45°

(c) 30°

(d) 90°

Let us first put the given data in the form of a diagram.

Let us draw radius OA, such that it touches the circle with center O at point A.

Let us draw radius O’B such that it touches the circle with center O’ at point B.

Let us also draw radii from each circle to the point C.

We know that the radius is always perpendicular to the tangent at the point of contact. Therefore,

That is,

…… (1)

Similarly,

Now, in ,

OA = OC(Radii of the same circle)

Therefore,

(Angles opposite to equal sides will be equal) …… (3)

Similarly, in

…… (4)

Consider the straight line OCO’. We have,

Looking at the figure, we can rewrite the above equation as,

(From (3) and (4))

(From (1) and (2))

…… (5)

Now let us take up. We have,

(Sum of all angles of a triangle will be 180°)

From equation (5), we get

Therefore the correct answer to this question is choice (d).

#### Question 8:

ABC is a right angled triangle, right angled at B such that BC = 6 cm and AB = 8 cm. A circle with centre O is inscribed in Δ ABC. The radius of the circle is

(a) 1 cm

(b) 2 cm

(c) 3 cm

(d) 4 cm

Let us first put the given data in the form of a diagram.

Let us first find out AC using Pythagoras theorem.

Also, we know that tangents drawn from an external point will be equal in length. Therefore we have the following,

BL = BM …… (1)

CM = CN …… (2)

AL = AN …… (3)

We have found that,

AC = 10

That is,

AN + NC = 10

But from (2) and (3), we can say

AL + MC = 10 …… (4)

It is given that,

AB = 8

BC = 6

Therefore,

AB + BC =14

Looking at the figure, we can rewrite this as,

AL + LB + BM + MC = 14

(AL + MC) + (BM + BL) = 14

Using (1) and (3) we can write the above equation as,

10 + 2BL = 14

BL = 2Consider the quadrilateral, BLOM, we have,

BL = BM (From (1))

OL = OM(Radii of the same circle)

(Given data)

(Radii is always perpendicular to the tangent at the point of contact)

(Radii is always perpendicular to the tangent at the point of contact)

Since the sum of all angles of a quadrilateral will be equal to , we have,

Since all the angles of the quadrilateral are equal to and since adjacent sides are equal, the quadrilateral BLOM is a square. We know that all the sides of a square are of equal length.

We have found BL = 2

Therefore,

OM = 2

OM is nothing but the radius of the circle.

Therefore, radius of the circle is 2 cm.

Hence the correct answer to the question is option (b).

#### Question 9:

PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such that ∠POR = 120°, then ∠OPQ is

(a) 60°

(b) 45°

(c) 30°

(d) 90°

Let us first put the given data in the form of a diagram.

Given data is as follows:

QOR is the diameter.

=

We have to find .

Since QOR is the diameter of the circle, it is a straight line. Therefore,

That is,

But = . Therefore,

Now consider . We have

(Sum of all angles of a triangle will be )

But,

(Since radius will be perpendicular to the tangent at the point of contact)

Therefore,

Therefore, the correct answer to this question is option (c).

#### Question 10:

If four sides of a quadrilateral ABCD are tangential to a circle, then

(a) AC + AD = BD + CD

(b) AB + CD = BC + AD

(c) AB + CD = AC + BC

(d) AC + AD = BC + DB

The figure of the Quadrilateral is drawn below.

We know that length of the tangents drawn from an external point will be equal. Therefore,

AP = PQ

DP = DS

CR = CS

BR = BQ

Let us add all the above four equations. We get

AP + DP + CR + BR = PQ + DS + CS + BQ

By looking at the figure, we can rewrite the above equation as,

AD + BC = AB + CD

Therefore, the correct answer is option (c).

#### Question 11:

The length of the tangent drawn from a point 8 cm away form the centre of a circle of radius 6 cm is

(a)

(b) $2\sqrt{7}$

(c) 10 cm

(d) 5 cm

Let us first put the given data in the form of a diagram.

We know that the radius of a circle will always be perpendicular to the tangent at the point of contact. Therefore, OP is perpendicular to QP. We can now use Pythagoras theorem to find the length of QP.

Therefore, the correct answer is option (b).

#### Question 12:

AB and CD are two common tangents to circles which touch each other at C. If D lies on AB such that CD =  4 cm, then AB is equal to

(a) 4 cm

(b) 6 cm

(c) 8 cm

(d) 12 cm

Let us first put the given data in the form of a diagram.

We know that tangents drawn from an external point will be equal.

Therefore,

Since CD is given as 4 cm,

Similarly,

BD = CD

Therefore,

BD = 4 cm

AB = 4 + 4

AB = 8 cm

Therefore, the answer to this question is option (c).

#### Question 13:

In the given figure, if AD, AE and BC are tangents to the circle at D, E and F respectively, Then,

(a) AD = AB + BC + CA

(b) 2AD = AB + BC + CA

(c) 3AD = AB + BC + CA

(d) 4AD = AB + BC + CA

In the given problem, the Right Hand Side of all the options is same, that is,

AB + BC + CA

So, we shall find out AB + BC + CA and check which of the options has the Left Hand Side value which we will arrive at.

By looking at the figure, we can write,

AB + BC + CA = AB + BF + FC + CD

We know that tangents drawn from an external point will be equal in length. Therefore,

BF = BE

FC= CD

Now we have,

AB + BC + CA = AB + BE + CD + CA

AB + BC + CD = (AB + BE) + (CD + CA)

By looking at the figure, we write the above equation as,

AB + BC + CD = AE + AD

Since tangents drawn from an external point will be equal,

Therefore,

AB + BC + CD = 2AD

Therefore option (b) is the correct answer.

#### Question 14:

In the given figure, RQ is a tangent to the circle with centre O. If SQ = 6 cm and QR = 4 cm, then OR =

(a) 8 cm

(b) 3 cm

(c) 2.5 cm

(d) 5 cm

It is given that,

SQ = 6 cm

Since SQ passes through the centre of the circle O, it is the diameter. Therefore, the radius,

OQ = 3 cm

Also, given is

QR = 4

We know that the radius will always be perpendicular to the tangent at the point of contact. Therefore, OQ is perpendicular to OR. We can find the length of OR by using Pythagoras theorem. We have,

Therefore option (d) is the correct answer.

#### Question 15:

In the given figure, the perimeter of Δ ABC is

(a) 30 cm

(b) 60 cm

(c) 45 cm

(d) 15 cm

We know that tangents from an external point will be equal in length.

Therefore,

AQ = AR

AQ is given as 4 cm. Therefore,

AR = 4

Similarly,

PC = CQ

PC = 5

Therefore,

CQ = 5

Similarly,

BP = BR

BR = 6

Therefore,

BP = 6

Now we can find out the perimeter of the triangle.

Perimeter = AB + BC + CA

From the figure we have,

Perimeter = AR + RB + BP + PC + CQ + QA

Perimeter = 4 + 6 + 6 + 5 + 5 + 4

Perimeter = 30 cm

Therefore, option (a) is the correct answer.

#### Question 16:

In the given figure, AP is a tangent to the circle with centre O such that OP = 4 cm and ∠OPA = 30°. Then,
AP =

(a)

(b) 2 cm

(c) $2\sqrt{3}$ cm

(d) $3\sqrt{2}cm$

In the given figure,

if we join O and A, the line OA will be perpendicular to AP. This is because the radius of a circle will always be perpendicular to the tangent at the point of contact. Therefore, is a right triangle.

We know that,

…… (1)

We know that,

…… (2)

From equations (1) and (2), we have,

Therefore, option (c) is the correct answer.

#### Question 17:

AP and PQ are tangents drawn from a point A to a circle with centre O and radius 9 cm. If OA = 15 cm, then AP + AQ =

(a) 12 cm

(b) 18 cm

(c) 24 cm

(d) 36 cm

Let us first put the given data in the form of a diagram.

We know that the radius of the circle will always be perpendicular to the tangent at the point of contact. Therefore,

Since tangents drawn from an external point will be equal in length,

AP = AQ

Since, AP = 12

AQ = 12

AP + AQ = 12 + 12

AP + AQ = 24

Therefore option (c) is the correct answer to this question.

#### Question 18:

At one end of a diameter PQ of a circle of radius 5 cm, tangent XPY is drawn to the circle. The length of chord AB parallel to XY and at a distance of 8 cm from P is

(a) 5 cm

(b) 6 cm

(c) 7 cm

(d) 8 cm

Consider the figure.
fIG

In right angled $∆\mathrm{OMB}$,

By pythagoras theorem.

${\mathrm{OB}}^{2}={\mathrm{MB}}^{2}+{\mathrm{OM}}^{2}\phantom{\rule{0ex}{0ex}}⇒{5}^{2}={\mathrm{MB}}^{2}+{3}^{2}\phantom{\rule{0ex}{0ex}}⇒25-9={\mathrm{MB}}^{2}\phantom{\rule{0ex}{0ex}}⇒16={\mathrm{MB}}^{2}\phantom{\rule{0ex}{0ex}}⇒4=\mathrm{MB}$
Therefore, AM = 2MB = 2 × 4 = 8 cm.
Hence, the correct answer is option (d).

#### Question 19:

If PT is tangent drawn from a point P to a circle touching it at T and O is the centre of the circle, then ∠OPT + ∠POT =

(a) 30°

(b) 60°

(c) 90°

(d) 180°

Let us first put the given data in the form of a diagram.

We know that the radius will always be perpendicular to the tangent at the point of contact. Therefore,

Consider . We know that sum of all angles of a triangle will be . Therefore,

Since , we have,

Choice (c) is the right answer.

#### Question 20:

In the adjacent figure, If AB = 12 cm, BC = 8 cm and AC = 10 cm, then AD =

(a) 5 cm

(b) 4 cm

(c) 6 cm

(d) 7 cm

By looking at the given figure

We can write the following equations:

We know that tangents form an external point will be of equal length. Therefore,

DB = BE …… (1)

Hence we have,

From the figure, we have

BE + EC = 8 …… (2)

Let us subtract equation (2) from equation (3). We get,

AD − EC = 4 …… (3)

Since tangents from an external point will be equal, we have,

EC = CF

Therefore, equation (3) becomes,

AF − CF = 4 …… (4)

From the figure we have,

AF + CF = 10 …… (5)

Adding equation (4) and (5), we get,

2AF = 14

AF = 7

We know that,

Therefore,

The correct answer is option (d).

#### Question 21:

In the given figure, if AP = PB, then

(a) AC = AB

(b) AC = BC

(c) AQ = QC

(d) AB = BC

The given figure is below

It is given that,

AP = PB

We know that

PB = BR(tangents from an external point are equal)

From the above two equations, we can say that,

AP = BR …… (1)

Also,

AP = AQ(tangents from an external point are equal) …… (2)

From equations (1) and (2), we have,

AQ = BR …… (3)

Also,

QC = CR(tangents from an external point are equal) …… (4)

Adding equations (3) and (4), we get,

AQ + QC = BR + CR

By looking at the figure we can write the above equation as,

AC = BC

The correct answer is option (b).

#### Question 22:

In the given figure, if AP = 10 cm, then BP =

(a) $\sqrt{91}cm$

(b) $\sqrt{127}cm$

(c) $\sqrt{119}cm$

(d) $\sqrt{109}cm$

Since the radius is always perpendicular to the tangent at the point of contact,

Therefore,

Now, consider. Here also, OB is perpendicular to PB since the radius will be perpendicular to the tangent at the point of contact. Therefore,

The correct answer is option (b).

#### Question 23:

In the given figure, if PR is tangent to the circle at P and Q is the centre of the circle, then ∠POQ =

(a) 110°

(b) 100°

(c) 120°

(d) 90°

We know that the radius is always perpendicular to the tangent at the point of contact.

Therefore, we have,

It is given that,

That is,

Now, consider. We have,

OP = OQ (Radii of the same circle)

Since angles opposite to equal side will be equal in a triangle, we have,

We know that sum of all angles of a triangle will be equal to .

Therefore,

The correct answer is option (c).

#### Question 24:

In the given figure, if quadrilateral PQRS circumscribes a circle, then PD + QB =

(a) PQ

(b) QR

(c) PR

(d) PS

We know that tangents drawn to a circle from the same external point will be equal in length.

Therefore,

PD = PA …… (1)

QB = QA …… (2)

Adding equations (1) and (2), we get,

PD + QB = PA + QA

By looking at the figure we can say,

PD + QB = PQ

Therefore option (a) is the correct answer.

#### Question 25:

In the given figure, two equal circles touch each other at T, if OP = 4.5 cm, then QR =

(a) 9 cm

(b) 18 cm

(c) 15 cm

(d) 13.5 cm

We know that tangents drawn from the same external point will be equal in length.

Therefore, we have,

QP = PT

PT = PR

From the above two equations, we get,

QP = PR

It is given that,

QP = 4.5 cm

Therefore,

PR = 4.5 cm

From the figure, we have,

QR = QP + PR

QR = 4.5 + 4.5

QR = 9

The correct answer is option (a).

#### Question 26:

In the given figure, APB is a tangent to a circle with centre O at point P. If ∠QPB = 50°, then the measure of ∠POQ is

(a) 100°

(b) 120°

(c) 140°

(d) 150°

We know that the radius of a circle will always be perpendicular to the tangent at the point of contact.

Therefore,

That is,

It is given that,

Therefore, we have,

Now, consider . We have,

OP = OQ(Radii of the same circle)

Since angles opposite to equal sides will be equal, we have,

We have found that,

Therefore,

We know that sum of all angles of a triangle will be equal to . Therefore,

The correct answer is option (a).

#### Question 27:

In the given figure, if tangents PA and PB are drawn to a circle such that ∠APB = 30° and chord AC is drawn parallel to the tangent PB, then ∠ABC =

(a) 60°

(b) 90°

(c) 30°

(d) None of these

We know that tangents from an external point will be equal in length. Therefore,

PA = PB

Now consider . We have,

PA = PB

We know that angles opposite to equal sides will be equal. Therefore,

Also, sum of all angles of a triangle will be equal to .

Now, as AC || PB;

Also,

The correct answer is option (c).

#### Question 28:

In the given figure, PR =

(a) 20 cm

(b) 26 cm

(c) 24 cm

(d) 28 cm

We know that the radius will always be perpendicular to the tangent at the point of contact.

Therefore,

Hence we have,

Also,

OO’ = sum of the radii of the two circles

OO’ = 3 + 5

OO’ = 8

Since radius is perpendicular to the tangent, . Therefore,

PR = PO + OO’ + O’R

PR = 5 + 8 + 13

PR =26

Therefore, option (b) is correct.

#### Question 29:

Two circles of same radii r and centres O and O' touch each other at P as shown in Fig. 10.91. If O O' is produced to meet the cirele C (O', r) at A and AT is a tangent to the circle C (O,r) such that O'Q $\perp$ AT. Then AO : AO' =

(a) 3/2

(b) 2

(c) 3

(d) 1/4

From the given figure we have,

AO = r + r + r

AO = 3r

AO’ = r

Therefore,

Also as therefore

Therefore, option (c) is correct.

#### Question 30:

Two concentric circles of radii 3 cm and 5 cm are given. Then length of chord BC which touches the inner circle at P is equal to

(a) 4 cm

(b) 6 cm

(c) 8 cm

(d) 10 cm

Consider .

We have,

Therefore,

Considering AB as the chord to the bigger circle, as OQ is perpendicular to AB, OQ bisects AB.

∴ AQ = QB = 4 cm.

Now, as BQ and BP are pair of tangents to the inner circle drawn from the external point B, QB = PB = 4 cm.

Now, join OP.

Then, .

⇒ OP bisects BC

⇒ BP = PC = 4 cm

Thus, BC = 4 cm + 4 cm = 8 cm

Therefore, option (c) is correct.

#### Question 31:

In the given figure, there are two concentric circles with centre O. PR and PQS are tangents to the inner circle from point plying on the outer circle. If PR = 7.5 cm, then PS is equal to

(a) 10 cm

(b) 12 cm

(c) 15 cm

(d) 18 cm

Let us first draw the radii OS, OP and OQ in the given figure, for our convenience.

Consider . We have,

PO = OS (Radii of the same circle)

Therefore, is an isosceles triangle.

We know that in an isosceles triangle, if a line is drawn perpendicular to the base of the triangle from the common vertex of the equal sides, then that line will bisect the base (unequal side).

Therefore, we have

PQ = QS

It is given that,

PQ = 7.5

Therefore,

QS = 7.5

From the figure, we have,

PS = PQ + QS

PS = 7.5 + 7.5

PS = 15

Therefore, option (c) is the correct answer.

#### Question 32:

In the given figure, if AB = 8 cm and PE = 3 cm, then AE =

(a) 11 cm

(b) 7 cm

(c) 5 cm

(d) 3 cm

We know that tangents drawn from the same external point will be equal in length.

Therefore,

AB = AC

It is given that,

AB = 8 cm

Hence,

AC = 8 cm …… (1)

Similarly,

PE = CE

It is given that,

PE = 3

Therefore,

CE = 3 …… (2)

Subtracting equations (1) and (2), we get,

AC − CE = 8 − 3

From the figure we can see that,

AC − CE = AE

Therefore,

AE = 8 − 3

AE = 5 cm

Option (c) is the correct answer.

#### Question 1:

Fill in the blanks:

(i) The common point of a tangent and the circle is called ..........

(ii) A circle may have .............. parallel tangents.

(iii) A tangent to a circle intersects in it ............ point(s).

(iv) A line intersecting a circle in two points is called a ...........

(v) The angle between tangent at a point on a circle and the radius through the point is ........

(i) We know that the tangent to a circle is that line which touches the circle at exactly one point. This point at which the tangent touches the circle is known as the ‘point of contact’.

Therefore we have,

The common point of the tangent and the circle is called Point of contact.

(ii) We know that the tangent is perpendicular to the radius of the circle at the point of contact. This also means that the tangent is perpendicular to the diameter of the circle at the point of contact. The diameter of the circle can have at the most one more perpendicular line at the other end where it touches the circle. The perpendicular line at the other end of the diameter is the tangent.

Also we know that two lines which are perpendicular to a common line will be parallel to each other.

Therefore,

A circle may have two parallel tangents.

(iii) From the very basic definition of tangent we know that tangent is a line that intersects the circle at exactly one point. Therefore we have,

A tangent to a circle intersects it in one point.

(iv) From the definition of a secant we know that any line that intersects the circle at 2 points is a secant. Therefore, we have

A line intersecting a circle in two points is called a secant.

(v) One of the properties of the tangent is that it is perpendicular to the radius at the point of contact. Therefore,

The angle between the tangent at the point of contact on a circle and the radius through the point is 90°.

#### Question 2:

How many tangents can a circle have?

We know that circle is made of infinite points which are located at a fixed distance from a particular point. Since at each of these infinite points a tangent can be drawn, we can have infinite number of tangent for a given circle.

#### Question 3:

O is the centre of a circle of a radius 8 cm. the tangent at a point A on the circle cuts a line through O at B such that AB = 15 cm. Find OB.

First let us draw whatever is given in the question. This will help us understand the problem better.

Since the tangent will always be perpendicular to the radius we have drawn OA perpendicular to AB. To find the length of OB we have to use Pythagoras theorem.

Therefore, length of OB is 17 cm.

#### Question 4:

If the tangent at a point P to a circle with centre O cuts a line through O at Q such that PQ = 24 cm and OQ = 25 cm. Find the radius of the circle.

Let us first draw whatever is given so that we can understand the problem better.

Since the tangent to a circle is always perpendicular to the radius of the circle at the point of contact, we have drawn OP perpendicular to PQ. Thus we have a right triangle with one of its sides as the radius. To find the radius we have to use Pythagoras theorem.

Therefore the radius of the circle is 7 cm.

#### Question 33:

In the given figure, PQ and PR are tangents drawn from P to a circle with centre O. If ∠OPQ = 35°, then

(a) a= 30°, b= 60°

(b) a= 35°, b = 55°

(c) a= 40°, b = 50°

(d) a= 45°, b = 45°

Consider and . We have,

PO is the common side for both the triangles.

OQ = OR(Radii of the same circle)

PQ = PR(Tangents from an external point will be equal)

Therefore, from SSS postulate of congruent triangles, we have,

Therefore,

That is,

It is given that,

Therefore,

Now consider . We know that the radius of a circle will always be perpendicular to the tangent at the point of contact. Therefore,

We know that sum of all angles of a triangle will always be equal to . Therefore,

Therefore, the correct answer is option (b).

#### Question 34:

In the given figure, if TP and TQ are tangents drawn from an external point T to a circle with centre O such that ∠TQP = 60°, then ∠OPQ =

(a) 25°

(b) 30°

(c) 40°

(d) 60°

Consider .

We have,

TP = TQ(Tangents from an external point will be equal)

We know that angles opposite to equal sides will be equal. Therefore,

It is given that,

Therefore,

We know that the radius will always be perpendicular to the tangent at the point of contact. Therefore,

Hence,

That is,

We have found that,

Therefore,

Therefore, the correct answer is option (b).

#### Question 35:

In the given figure, the sides AB , BC and CA of triangle ABC, touch a circle at P, Q and R respectively. If PA = 4 cm, BP = 3 cm and AC = 11 cm, then length of BC is

(a) 11 cm                          (b) 10 cm                          (c) 14 cm                          (d) 15 cm                           [CBSE 2012]

It is given that the sides AB , BC and CA of ∆ABC touch a circle at P, Q and R, respectively.

Also, PA = 4 cm, PB = 3 cm and AC = 11 cm

We know that, the lengths of tangents drawn from an external point to a circle are equal.

∴ AR = AP = 4 cm

BQ = BP = 3 cm

Now, CR = AC − AR = 11 cm − 4 cm = 7 cm

∴ CQ = CR = 7 cm         (Lengths of tangents drawn from an external point to a circle are equal)

Now,

BC = BQ + CQ = 3 cm + 7 cm = 10 cm

Thus, the length of BC is 10 cm.

Hence, the correct answer is option B.

#### Question 36:

In the given figure, a circle touches the side DF of ΔEDF at H and touches ED and EF produced at K and M respectively. If EK = 9 cm, then the perimeter of ΔEDF is

(a) 18 cm                         (b) 13.5 cm                         (c) 12 cm                         (d) 9 cm                              [CBSE 2012]

In the given figure, DH and DK are tangents drawn to the circle from an external point D.

∴ DH = DK                      (Lengths of tangents drawn from an external point to a circle are equal)

FH and FM are tangents drawn to the circle from an external point F.

∴ FH = FM                      (Lengths of tangents drawn from an external point to a circle are equal)

EK and EM are tangents drawn to the circle from an external point E.

∴ EM = EK = 9 cm          (Lengths of tangents drawn from an external point to a circle are equal)

Now,

Perimeter of ΔEDF = ED + DF + EF

= ED + (DH + FH) + EF

= ED + (DK + FM) + EF                (DH = DK and FH = FM)

= (ED + DK) + (FM + EF)

= EK + EM

= 9 cm + 9 cm

= 18 cm

Thus, the perimeter of ΔEDF is 18 cm.

Hence, the corrrect answer is option A.

#### Question 37:

In the given figure, DE and DF are tangents from an external point D to a circle with centre A. If DE = 5 cm and DE ⊥ DF, then the radius of the circle is

(a) 3 cm                             (b) 5 cm                             (c) 4 cm                             (d) 6 cm                             [CBSE 2013]

It is given that DE and DF are tangents from an external point D to a circle with centre A. DE = 5 cm and DE ⊥ DF.

Join AE and AF.

Now, DE is a tangent at E and AE is the radius through the point of contact E.

$\therefore \angle \mathrm{AED}=90°$     (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

Also, DF is a tangent at F and AF is the radius through the point of contact F.

$\therefore \angle \mathrm{AFD}=90°$     (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

$\angle \mathrm{EDF}=90°$         (DE ⊥ DF)

Also, DF = DE        (Lengths of tangents drawn from an external point to a circle are equal)

So, AEDF is a square.

∴ AE = AF = DE = 5 cm         (Sides of square are equal)

Thus, the radius of the circle is 5 cm.

Hence, the correct answer is option B.

#### Question 38:

In the given figure, a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches sides BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29 cm, AD = 23 cm, ∠B = 90° and DS = 5 cm, then the radius of the circle (in cm) is

(a) 11                            (b) 18                            (c) 6                            (d) 15                           [CBSE 2013]

It is given that the sides BC, AB, AD and CD touches a circle with centre O at points P, Q, R and S, respectively.

AB = 29 cm, AD = 23 cm, ∠B = 90° and DS = 5 cm.

Now,

DR = DS = 5 cm           (Lengths of tangents drawn from an external point to a circle are equal)

AR = AD − DR = 23 cm − 5 cm = 18 cm

∴ AQ = AR = 18 cm           (Lengths of tangents drawn from an external point to a circle are equal)

BQ = AB − AQ = 29 cm − 18 cm = 11 cm

∴ BP = BQ = 11 cm           (Lengths of tangents drawn from an external point to a circle are equal)

Now, AB is a tangent at Q and OQ is the radius through the point of contact Q.

$\therefore \angle \mathrm{OQB}=90°$     (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

BC is a tangent at P and OP is the radius through the point of contact P.

$\therefore \angle \mathrm{OPB}=90°$     (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

$\angle \mathrm{B}=90°$      (Given)

Also, BP = BQ

So, OPBQ is a square.

∴ OP = OQ = BQ = 11 cm         (Sides of square are equal)

Thus, the radius of the circle is 11 cm.

Hence, the correct answer is option A.

#### Question 39:

In a right triangle ABC, right angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle (in cm) is

(a) 4                                    (b) 3                                    (c) 2                                    (d) 1                                   [CBSE 2014]

Let a circle with centre O and radius r cm be inscribed in the right ∆ABC.

In right ∆ABC,

Now, the sides AB, BC and CA are tangents to the circle at R, P and Q, respectively.

∴ OR ⊥ AB, OP ⊥ BC and OQ ⊥ AC

Now,

ar(∆OAB) + ar(∆OBC) + ar(∆OAC) = ar(∆ABC)

$⇒\frac{1}{2}×\mathrm{AB}×\mathrm{OR}+\frac{1}{2}×\mathrm{BC}×\mathrm{OP}+\frac{1}{2}×\mathrm{AC}×\mathrm{OQ}=\frac{1}{2}×\mathrm{BC}×\mathrm{AB}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}×5×r+\frac{1}{2}×12×r+\frac{1}{2}×13×r=\frac{1}{2}×12×5\phantom{\rule{0ex}{0ex}}⇒5r+12r+13r=60\phantom{\rule{0ex}{0ex}}⇒30r=60\phantom{\rule{0ex}{0ex}}⇒r=2$

Thus, the radius of the circle is 2 cm.

Hence, the correct answer is option C.

#### Question 40:

Two circles touch each other externally at P. AB is a common tangent to the circle touching them at A and B. The value of $\angle$APB is

(a) 30º                             (b) 45º                             (c) 60º                             (d) 90º                                       [CBSE 2014]

It is given that two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B.

Draw a tangent to the circles at P, intersecting AB at T.

Now, TA and TP are tangents drawn to the same circle from an external point T.

∴ TA = TP            (Lengths of tangents drawn from an external point to a circle are equal)

TB and TP are tangents drawn to the same circle from an external point T.

∴ TB = TP            (Lengths of tangents drawn from an external point to a circle are equal)

In ∆ATP,

TA = TP

$\therefore \angle \mathrm{APT}=\angle \mathrm{PAT}$      .....(1)           (In a triangle, equal sides have equal angles opposite to them)

In ∆BTP,

TB = TP

$\therefore \angle \mathrm{BPT}=\angle \mathrm{PBT}$      .....(2)            (In a triangle, equal sides have equal angles opposite to them)

Now, in ∆APB,

$\angle \mathrm{APB}+\angle \mathrm{PAB}+\angle \mathrm{PBA}=180°$       (Angle sum property)

Thus, the value of $\angle$APB is 90º.

Hence, the correct answer is option D.

#### Question 41:

In the given figure, PQ and PR are two tangents to a circle with centre O. If $\angle$QPR = 46º, then $\angle$QOR is

(a) 67º                             (b) 134º                             (c) 44º                             (d) 46º                                       [CBSE 2014]

It is given that PQ and PR are two tangents to a circle with centre O.

Now, PQ is a tangent at Q and OQ is the radius through the point of contact Q.

$\angle$OQP = 90º               (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

PR is a tangent at R and OR is the radius through the point of contact R.

$\angle$ORP = 90º               (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

Also, $\angle$QPR = 46º    (Given)

$\angle$OQP + $\angle$QPR + $\angle$ORP + $\angle$QOR = 360º                  (Angle sum property)

⇒ 90º + 46º + 90º + $\angle$QOR = 360º

⇒ 226º + $\angle$QOR = 360º

$\angle$QOR = 360º − 226º = 134º

$\angle$QOR = 134º

Hence, the correct answer is option B.

#### Question 42:

In the given figure, QR is a common tangent to the given circles touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then the length of QR (in cm) is

(a) 3.8                                 (b) 7.6                                (c) 5.7                                 (d) 1.9                            [CBSE 2014]

It is given that QR is a common tangent to the given circles touching externally at the point T. Also, the tangent at T meets QR at P such that PT = 3.8 cm.

Now, PQ and PT are tangents drawn to the same circle from an external point P.

∴ PQ = PT = 3.8 cm            (Lengths of tangents drawn from an external point to a circle are equal)

PR and PT are tangents drawn to the same circle from an external point T.

∴ PR = PT = 3.8 cm            (Lengths of tangents drawn from an external point to a circle are equal)

Now,

QR = PQ + PR = 3.8 cm + 3.8 cm = 7.6 cm

Thus, the length of QR is 7.6 cm.

Hence, the correct answer is option B.

#### Question 43:

In the given figure, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively. If AB = x cm, BC = 7 cm, CR = 3 cm and AS = 5 cm, then x =

(a) 10                                (b) 9                                (c) 8                                (d) 7                                            [CBSE 2014]

It is given that a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S, respectively.

Also, BC = 7 cm, CR = 3 cm and AS = 5 cm

Now, AP and AS are tangents drawn to the same circle from an external point A.

∴ AP = AS = 5 cm          .....(1)                (Lengths of tangents drawn from an external point to a circle are equal)

CQ and CR are tangents drawn to the same circle from an external point C.

∴ CQ = CR = 3 cm           (Lengths of tangents drawn from an external point to a circle are equal)

So, BQ = BC − CQ = 7 cm − 3 cm = 4 cm

BP and BQ are tangents drawn to the same circle from an external point B.

BP = BQ = 4 cm          .....(2)                    (Lengths of tangents drawn from an external point to a circle are equal)

Now,

AB = AP + PB = 5 cm + 4 cm = 9 cm            [From (1) and (2)]

x = 9              (AB = x cm)

Hence, the correct answer is option B.

#### Question 44:

If angle between two radii of a circle is 1300 , the angle between the tangents at the ends of radii is
(a)   900       (b)    500       (c)    700        (d)   400

Let PQ and RP be the radii of the circle with the centre O.
$\angle \mathrm{ROQ}=130°$
(Radii are perpendicular to the tangent)
$\angle \mathrm{ORP}+\angle \mathrm{RPQ}+\angle \mathrm{PQO}+\angle \mathrm{QOR}=360°\phantom{\rule{0ex}{0ex}}⇒90°+\angle \mathrm{RPQ}+90°+130°=360°\phantom{\rule{0ex}{0ex}}⇒\angle \mathrm{RPQ}=50°$
Hence, the correct answer is option (b).

#### Question 45:

If two tangents inclined at an angle  of 60 are drawn to a circle of radius 3 cm , then length of each tangent is equal to
(a)  $\frac{3\sqrt{3}}{2}$    (b)   6 cm     (c)    3 cm    (d)    $3\sqrt{3}$ cm

Let PA and PB be two tangents to a circle with centre O and radius 3 cm.

We are given ∠APB = 60°

We know that two tangents drawn to a circle from an external point are equally inclined to the segment joining the centre to the point.

∴ ∠APO = ∠BPO =  $\frac{1}{2}$ × ∠APB = $\frac{1}{2}$ × 60° = 30°

Also, OA ⊥ AP and OB ⊥ BP (radius  ⊥ tangent at point of contact)

In right ΔOAP,
$\mathrm{tan}30°=\frac{3}{PA}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{\sqrt{3}}=\frac{3}{PA}\phantom{\rule{0ex}{0ex}}⇒PA=3\sqrt{3}$

Therefore, PA = PB =
Hence, the correct answer is option (d).

#### Question 46:

If radii of two concentric circles are 4 cm and 5 cm , then the length of each chord of one circle which is tangent to the other circle is
(a) 3 cm         (b)   6 cm         (c)   9 cm         (d)   1 cm

Let the chord of one circle which tangent to the other be AB.
In ∆AOC,

thus, AB = AC + CB = 3 + 3 = 6 cm
Hence, the correct answer is option (b).

#### Question 47:

At one end A of a diameter AB of a circle of radius 5 cm , tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A is
(a) 4 cm    (b)  5 cm    (c)    6 cm     (d)   8 cm

XY is the tangent to the circle with centre O.
CD is the chord.
OA = OB = OD = 5 cm                            (Radii)
PA = 8 cm
PO = 3 cm
In ∆POD,

Hence, CD = CP + PD = 4 + 4 = 8 cm
so, the correct answer is option (d).

#### Question 48:

From a point P which is at a distance 13 cm from the centre O of a circle of radious  5 cm , the pair of tangents PQ and PR to the circle are drawn . Then the area of the quadrilateral PQOR is
(a)   60 cm   (b)  65 cm2     (c)   30 cm   (d)  32.5 cm2

Clearly tangent PQ and PR are perpendicular to OQ and OR respectively.
Hence both triangles POQ and PQR are right angled.

Area of ∆POQ =

Similarly,
Area of ∆POR = Area of ∆POQ = 30 cm2                     (Both the triangles are symmetrical)
Area of quadrilateral PQOR = 30 + 30 = 60 cm2
Hence, the correct answer is option (a).

#### Question 49:

If PA and PB are tangents to the circle with centre O such that $\angle$APB = 500, then $\angle$OAB is equal to

(a) 250    (b)  300    (c) 40 (d) 500

$\angle APB=50°$
$\angle OAP+\angle APB+\angle PBO+\angle AOB=360°\phantom{\rule{0ex}{0ex}}⇒90°+90°+50°+\angle AOB=360°\phantom{\rule{0ex}{0ex}}⇒\angle AOB=130°$
In ∆AOB,
Let $\angle OAB=\angle OBA=x$            (OA = OB as they are radii)
$\angle AOB+\angle OBA+\angle OAB=180°\phantom{\rule{0ex}{0ex}}⇒130°+2x=180\phantom{\rule{0ex}{0ex}}⇒x=25°$
Hence, the correct answer is option (a).

#### Question 50:

The pair of tangents AP and AQ drawn from an external point to a circle with centre O are perpendicular to each other and length of each tangent is 5 cm. The radius of the circle is
(a) 10 cm    (b)  7.5 cm    (c)   5cm     (d)   2.5 cm

Given:
AP and AQ are tangents to the circle with centre O, AP ⊥ AQ
and AP = AQ = 5 cm
we know that radius of a circle is perpendicular to the tangent at the point of contact
OP ⊥ AP and OQ ⊥ AQ
Also sum of all angles of a quadrilateral is 360°

⇒∠O + ∠P + ∠A + ∠Q = 360°

⇒∠O + 90° + 90° + 90° = 360°

⇒∠O = 360° – 270° = 90°

Thus ∠O = ∠P = ∠A = ∠Q = 90°
OPAQ is a rectangle
Since adjacent sides of OPAQ i.e. AP and AQ are equal. Thus OPAQ is a square
radius = OP = OQ = AP = AQ = 5 cm
Hence, the correct answer is option (c).

#### Question 51:

In Fig.10.114, if  , then $\angle COD$ is equal to
figure

(a) 450      (b) 350    (c)   55    (d)  $62{1}{2}}^{°}$

We know that opposite sides of a quadrilateral circumscribing a circle subtend supplementary
angles at the centre.
Quadrilateral ABCD circumscribe a circle with centre O.
$\angle AOB+\angle COD=180°\phantom{\rule{0ex}{0ex}}⇒125°+\angle COD=180°\phantom{\rule{0ex}{0ex}}⇒\angle COD=180°-125°\phantom{\rule{0ex}{0ex}}⇒\angle COD=55°$
Hence, the correct answer is option (c).

#### Question 52:

In Fig . 10.115, if PQR  is the tangent to a circle at Q whose centre is O , AB is a chord parallel to PR and , then $\angle AQB$ is equal to
figure

(a) 200       (b) 40   9c) 350   (d) 450
0

Given: PQR is the tangent to the circle with centre O.
$\angle \mathrm{BQR}=70°$

$\angle \mathrm{BSQ}=90°$                                    (radius is perpendicular to the chord)
In ∆BSQ,
$70°+\angle \mathrm{BQS}+90°=180°\phantom{\rule{0ex}{0ex}}⇒\angle \mathrm{BQS}=20°$
Similarly, $\angle \mathrm{AQS}=20°$
Thus,
$\angle \mathrm{BQS}+\angle \mathrm{AQS}=\angle \mathrm{AQB}\phantom{\rule{0ex}{0ex}}⇒\angle \mathrm{AQB}=20°+20°=40°$
Hence, the correct answer is option (b).

#### Question 1:

A line intersecting a circle in two distinct points is called a __________.

A line intersecting a circle in two distinct points is called a _secant_.

#### Question 2:

The tangents drawn, at the end-points of a diameter, to a circle are ___________.

AB is the diameter of the circle with centre O. CD and EF are the tangents to the circle at points A and B, respectively.

Now,

∠OAD = 90º                   (Radius is perpendicular to the tangent at the point of contact)

Also, ∠OBF = 90º          (Radius is perpendicular to the tangent at the point of contact)

∴ ∠OAD + ∠OBF = 90º + 90º = 180º

⇒ CD || EF      (If a pair of consecutive interior angles is supplementary, then the two lines are parallel)

Hence, the tangents drawn at the end-points of a diameter to a circle are parallel.

The tangents drawn, at the end-points of a diameter, to a circle are __parallel__.

#### Question 3:

The lengths of two tangents drawn from an external point to a circle are _________.

PA and PB are two tangents drawn from P to circle with centre O. OP is the line segment joining the point P and centre of the circle.

In ∆OAP and ∆OBP,

OA = OB      (Radius of the circle)

OP = OP       (Common)

∠OAP = ∠OBP    (Radius is perpendicular to the tangent at the point of contact)

∴ ∆OAP $\cong$ ∆OBP      (RHS congruence criterion)

⇒ PA = PB        (CPCT)

Hence, the lengths of two tangents drawn from an external point to a circle are equal.

The lengths of two tangents drawn from an external point to a circle are __equal__.

#### Question 4:

The two tangents drawn from an external point to a circle are _______ to the segment joining the centre to the point.

PA and PB are two tangents drawn from P to circle with centre O. OP is the line segment joining the point P and centre of the circle.

In ∆OAP and ∆OBP,

OA = OB      (Radius of the circle)

OP = OP       (Common)

∠OAP = ∠OBP    (Radius is perpendicular to the tangent at the point of contact)

∴ ∆OAP $\cong$ ∆OBP      (RHS congruence criterion)

⇒ ∠APO = ∠BPO      (CPCT)

Hence, the two tangents drawn from an external point to a circle are equally inclined to the segment joining the centre to the point.

The two tangents drawn from an external point to a circle are _equally inclined_ to the segment joining the centre to the point.

#### Question 5:

Tangents drawn from an external point to a circle subtend ________ angles to the centre.

PA and PB are two tangents drawn from P to circle with centre O. OP is the line segment joining the point P and centre of the circle.

In ∆OAP and ∆OBP,

OA = OB      (Radius of the circle)

OP = OP       (Common)

∠OAP = ∠OBP    (Radius is perpendicular to the tangent at the point of contact)

∴ ∆OAP $\cong$ ∆OBP      (RHS congruence criterion)

⇒ ∠AOP = ∠BOP      (CPCT)

Hence, the tangents drawn from an external point to a circle subtend equal angles to the centre.

Tangents drawn from an external point to a circle subtend __equal__ angles to the centre.

#### Question 6:

Parallelogram circumscribing a circle is a _______.

ABCD is a parallelogram such that its sides touches the circle with centre O at P, Q, R and S.

We know that the opposite sides of the parallelogram are equal. Therefore,

AB = CD     .....(1)

Also, the length of the tangents drawn from an external point to a circle are equal.

∴ AR = AS      .....(3)

DR = DQ         .....(4)

BP = SB          .....(5)

CP = CQ         .....(6)

Adding (3), (4), (5) and (6), we get

AR + DR + BP + CP = AS + SB + CQ + DQ

⇒ AD + BC = AB + CD

⇒ 2AD = 2AB          [From (1) and (2)]

From (1), (2) and (7), we have

AB = BC = CD = AD

Hence, the parallelogram ABCD is a rhombus.

Parallelogram circumscribing a circle is a _rhombus_.

#### Question 7:

If AB and CD are common tangents to two circles of unequal radii, then _________.

Here, AB and CD are common tangents to two circles of unequal radii and having centres O1 and O2

AB and CD are produced to intersect at P.

Now, PB and PD are tangents drawn from external point P to circle with centre O2.

∴ PB = PD     .....(1)                (Lengths of tangents drawn from an external point to a circle are equal)

Also, PA and PC are tangents drawn from external point P to circle with centre O1.

∴ PA = PC      .....(2)               (Lengths of tangents drawn from an external point to a circle are equal)

Subtracting (1) from (2), we get

PA − PB = PC − PD

⇒ AB = CD

If AB and CD are common tangents to two circles of unequal radii, then __AB = CD__.

#### Question 8:

From an external point P, two tangents PA and PB are drawn to the circle with centre O. Then the angle between OP and AB is_______.

PA and PB are tangents drawn from P to circle with centre O. Let OP and AB intersect at Q.

In ∆PAQ and ∆PBQ,

AP = BP                   (Length of tangents drawn from an external point to a circle are equal)

∠APQ = ∠BPQ       (Tangents drawn from an external point to a circle are equally inclined to the segment joining the centre to that point)

PQ = PQ                  (Common)

∴ ∆PAQ $\cong$ ∆PBQ      (SAS congruence axiom)

⇒ ∠AQP = ∠BQP      (CPCT)

Also,

∠AQP + ∠BQP = 180º       (Linear pair of angles)

⇒ 2∠AQP = 180º               (∠AQP = ∠BQP)

⇒ ∠AQP = 90º

Therefore, OP is perpendicular to AB i.e. the angle between OP and AB is 90º.

From an external point P, two tangents PA and PB are drawn to the circle with centre O. Then the angle between OP and AB is __90º__.

#### Question 10:

AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = ________.

AB is a diameter of circle with centre O and AC is the chord of the circle such that ∠BAC = 30°. The tangent at C intersects AB extended at D.

In ∆OAC,

OA = OC    (Radius of circle)

∴ ∠OCA = ∠OAC       (In a triangle, equal sides have equal angles opposite to them)

⇒  ∠OCA = 30°

Now, ∠OCD = 90°    (Radius is perpendicular to the tangent at the point of contact)

∴  ∠ACD  = ∠OCA + ∠OCD = 30° + 90° = 120°

In ∆ACD,

∠A + ∠ACD + ∠D = 180º       (Angle sum property)

⇒ 30° + 120° + ∠D = 180º

⇒ ∠D = 180º − 150° = 30°     .....(1)

Now, ∠ACB = 90°      (Angle in a semi-circle is 90°)

∠ACD = ∠ACB + ∠BCD

⇒ 120° = 90° + ∠BCD

⇒ ∠BCD = 120º − 90° = 30°         .....(2)

In ∆BCD,

∠D = ∠BCD    [From (1) and (2)]

⇒ BC = BD        (In a triangle, equal angles have equal sides opposite to them)

AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = _  BD _.

#### Question 11:

If a chord AB subtends an angle of 60° at the centre of a circle, then the angle between the tangents at A and B is ________.

AB is the chord of a circle with centre O such that ∠AOB = 60º. The tangents at A and B intersect at P.

Now,

∠OAP = 90º                   (Radius is perpendicular to the tangent at the point of contact)

Also, ∠OBP = 90º          (Radius is perpendicular to the tangent at the point of contact)

∠AOB + ∠OAP + ∠OBP + ∠APB = 360º         (Angle sum property of quadrilateral)

⇒ 60º + 90º + 90º + ∠APB = 360º

⇒ ∠APB = 360º − 240º = 120º

Thus, the angle between the tangents at A and B is 120º.

If a chord AB subtends an angle of 60° at the centre of a circle, then the angle between the tangents at A and B is __120º__.

#### Question 12:

The length of the tangent from an external point P on the circle with centre O is always ________ OP.

PT is a tangent drawn from an external point P on the circle with centre O.

∠OTP = 90º         (Radius is perpendicular to the tangent at the point of contact)

So, ∆OPT is a right triangle. OP is the hypotenuse of the right ∆OPT.

∴ OP > PT               (In a right triangle, hypotenuse is the longest side.)

Or PT < OP

Thus, the length of the tangent from an external point P on the circle with centre O is always less than the hypotenuse OP.

The length of the tangent from an external point P on the circle with centre O is always __less than__ OP.

#### Question 13:

If the angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = _______.

PA and PB are tangents drawn from P to circle with centre O.

∠APB = 60º and OA = OB = a  (Radius of the circle)

We know that, the two tangents drawn from an external point to a circle are equally inclined to the segment joining the centre to the point.

∴ ∠APO = ∠BPO = $\frac{60°}{2}$ = 30º

Also, ∠OAP = 90º       (Radius is perpendicular to the tangent at the point of contact)

In right ∆OAP,

$\mathrm{sin}30°=\frac{\mathrm{OA}}{\mathrm{OP}}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}=\frac{a}{\mathrm{OP}}\phantom{\rule{0ex}{0ex}}⇒\mathrm{OP}=2a$

If the angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = ___2a___.

#### Question 14:

If a number of circles pass through the end points P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of _______.

Consider two circles with centres O1 and O2 passing through the end points P and Q of a line segment PQ.

We know that the perpendicular bisector of the chord of a circle passes through the centre of the circle.

Thus, the perpendicular bisector PQ passes through the centres O1 and O2.

In the same manner, all circles passing through the end points P and Q of a line segment PQ will have their centres lying on the perpendicular bisector of PQ.

If a number of circles pass through the end points P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of ___PQ___.

#### Question 15:

The angle between two tangents drawn from an external point to a circle is _______ to the angle subtended by the line segments joining the points of contact at the centre.

PA and PB are two tangents drawn from P to circle with centre O.

Now,

∠OAP = 90º                   (Radius is perpendicular to the tangent at the point of contact)

Also, ∠OBP = 90º          (Radius is perpendicular to the tangent at the point of contact)

∠AOB + ∠OAP + ∠OBP + ∠APB = 360º         (Angle sum property of quadrilateral)

⇒ ∠AOB + 90º + 90º + ∠APB = 360º

⇒ ∠AOB + ∠APB = 360º − 180º = 180º

Thus, the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre.

The angle between two tangents drawn from an external point to a circle is __supplementary__ to the angle subtended by the line segments joining the points of contact at the centre.

#### Question 16:

PA and PB are two tangents drawn from an external point P to a circle with centre O. If ∠PBA = 65°, then ∠APB = _______.

PA and PB are two tangents drawn from an external point P to a circle with centre O.

∴ PA = PB      (Lengths of tangents drawn from an external point to a circle are equal)

In ∆APB,

PB = PA        (Proved)

⇒ ∠PAB = ∠PBA       (In a triangle, equal sides have equal angles opposite to them)

⇒ ∠PAB = 65°            (∠PBA = 65°)

Also,

∠PAB + ∠PBA + ∠APB = 180°        (Angle sum property of triangle)

⇒ 65° + 65° + ∠APB = 180°

⇒ ∠APB = 180° − 130° = 50°

PA and PB are two tangents drawn from an external point P to a circle with centre O. If ∠PBA = 65°, then ∠APB = ___50°___.

#### Question 17:

PA and PB are two tangents drawn from an external point P to a circle with centre O. If ∠APB = 80°, then ∠POA = __________.

PA and PB are tangents drawn from P to circle with centre O.

∠APB = 80º

We know that, the two tangents drawn from an external point to a circle are equally inclined to the segment joining the centre to the point.

∴ ∠APO = ∠BPO = $\frac{80°}{2}$ = 40º

Also, ∠OAP = 90º       (Radius is perpendicular to the tangent at the point of contact)

In ∆OAP,

∠APO + ∠OAP + ∠POA = 180º         (Angle sum property of triangle)

⇒ 40º + 90º + ∠POA = 180º

⇒ ∠POA = 180º − 130º = 50º

PA and PB are two tangents drawn from an external point P to a circle with centre O. If ∠APB = 80°, then ∠POA = ____50°____.

#### Question 18:

If PA and PB are two tangents drawn from an external point P to a circle such that PA = 5 cm and ∠APB = 60°, then the length of chord AB is ________.

PA and PB are tangents drawn from P to circle with centre O. Let OP and AB intersect at Q.

∠APQ = ∠BPQ = $\frac{60°}{2}$ = 30º       (Tangents drawn from an external point to a circle are equally inclined to the segment joining the centre to that point)

In ∆PAQ and ∆PBQ,

AP = BP                   (Length of tangents drawn from an external point to a circle are equal)

∠APQ = ∠BPQ       (30º each)

PQ = PQ                  (Common)

∴ ∆PAQ $\cong$ ∆PBQ      (SAS congruence axiom)

⇒ ∠AQP = ∠BQP and AQ = BQ      (CPCT)

Also,

∠AQP + ∠BQP = 180º       (Linear pair of angles)

⇒ 2∠AQP = 180º               (∠AQP = ∠BQP)

⇒ ∠AQP = 90º

In right ∆APQ,

∴ AB = 2AQ = $2×\frac{5}{2}$ = 5 cm      (AQ = BQ)

Thus, the length of the chord AB is 5 cm.

If PA and PB are two tangents drawn from an external point P to a circle such that PA = 5 cm and ∠APB = 60°, then the length of chord AB is ___5 cm___.

#### Question 19:

Quadrilateral ABCD is circumscribed to a circle. If AB = 6 cm, BC = 7 cm and CD = 4 cm, then AD = ________.

ABCD is a quadrilateral such that its sides touches the circle with centre O at P, Q, R and S.

We know that the lengths of the tangents drawn from an external point to a circle are equal. Therefore,

AR = AS          .....(1)

DR = DQ         .....(2)

BP = SB          .....(3)

CP = CQ         .....(4)

Adding (1), (2), (3) and (4), we get

AR + DR + BP + CP = AS + SB + CQ + DQ

⇒ AD + BC = AB + CD

⇒ AD + 7 cm = 6 cm + 4 cm

⇒ AD = 10 − 7 = 3 cm

Quadrilateral ABCD is circumscribed to a circle. If AB = 6 cm, BC = 7 cm and CD = 4 cm, then AD = __3 cm__.

#### Question 20:

The length of the tangent from an external point P to a circle of radius 5 cm is 10 cm. The distance of the point from the centre of the circle is _______.

PT is a tangents drawn from P to circle with centre O.

PT = 10 cm and OT = 5 cm   (Radius of the circle)

Now,

∠OTP = 90º       (Radius is perpendicular to the tangent at the point of contact)

In right ∆OTP,

Thus, the distance of the point from the centre of the circle is $5\sqrt{5}$ cm.

The length of the tangent from an external point P to a circle of radius 5 cm is 10 cm. The distance of the point from the centre of the circle is .

#### Question 1:

In the given figure, PA and PB are tangents to the circle drawn from an external point P. CD is a third tangent touching the circle at Q. If PB = 10 cm and CQ = 2 cm, what is the length PC?

Given data is as follows:

PB = 10 cm

CQ = 2 cm

We have to find the length of PC.

We know that the length of two tangents drawn from the same external point will equal. Therefore,

PB = PA

It is given that PB = 10 cm

Therefore, PA = 10 cm

Also, from the same principle we have,

CQ = CA

It is given that CQ = 2 cm

Therefore, CA = 2cm

From the given figure we can say that,

PC = PA − CA

Now that we know the values of PA and CA, let us substitute the values in the above equation.

PC = 10 − 2

PC = 8 cm

Therefore, length of PC is 8 cm.

#### Question 2:

What is the distance between two parallel tangents of a circle of radius 4 cm?

Two parallel tangents can exist at the two ends of the diameter of the circle. Therefore, the distance between the two parallel tangents will be equal to the diameter of the circle. In the problem the radius of the circle is given as 4 cm. Therefore,

Diameter =

Diameter = 8 cm

Hence, the distance between the two parallel tangents is 8 cm.

#### Question 3:

The length of tangent from a point A at a distance of 5 cm from the centre of the circle is 4 cm. What is the radius of the circle?

Let us first draw whatever is given for a better understanding of the problem.

Let O be the center of the circle and B be the point of contact.

We know that the radius of the circle will be perpendicular to the tangent at the point of contact. Therefore, we have as a right triangle and we have to apply Pythagoras theorem to find the radius of the triangle.

Therefore, radius of the circle is 3 cm.

#### Question 4:

Two tangents TP and TQ are drawn from an external point T to a circle with centre O as shown in Fig. 10.73. If they are inclined to each other at an angle of 100°, then what is the value of ∠POQ?

Consider the quadrilateral OPTQ. It is given that PTQ = 100°.

From the property of the tangent we know that the tangent will always be perpendicular to the radius at the point of contact. Therefore we have,

We know that the sum of all angles of a quadrilateral will always be equal to 360°.

Therefore,

+ ++

Let us substitute the values of all the known angles. We have,

Therefore, the value of angle POQ is 80°.

#### Question 5:

What the distance between two parallel tangents to a circle of radius 5 cm?

Two parallel tangents can exist at the two ends of the diameter of the circle. Therefore, the distance between the two parallel tangents will be equal to the diameter of the circle. In the problem the radius of the circle is given as 5 cm. Therefore,

Diameter = 5 × 2

Diameter = 10 cm

Hence, the distance between the two parallel tangents is 10 cm.

#### Question 6:

In Q.No. 1, if PB = 10 cm, what is the perimeter of Δ PCD?

Here, we have to find the perimeter of triangle PCD.

Perimeter is nothing but sum of all sides of the triangle. Therefore we have,

Perimeter of =

In the given figure we can see that,

=

Therefore,

Perimeter of =

We know that the two tangents drawn to a circle from a common external point will be equal in length. From this property we have,

Now let us replace CQ and QD with CA and DA. We get,

Perimeter of =

Also from the figure we can see that,

Now, let us replace these in the equation for perimeter of. We have,

Perimeter of = PB +PA

Also, from the property of tangents we know that, two tangents drawn to a circle from the same external point will be equal in length. Therefore,

PB = PA

Let us replace PA with PB in the above equation. We get,

Perimeter of = 2PB

It is given in the question that PB = 10 cm. Therefore,

Perimeter of =

Perimeter of = 20 cm

Hence, the perimeter of is 20 cm.

#### Question 7:

In the given figure, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm and BC = 7 cm, then find the length of BR.

CP and CQ are tangents drawn from an external point C to the circle.

∴ CP = CQ (Length of tangents drawn from an external point to the circle are equal)

⇒ CQ = 11 cm (CP = 11 cm)

BQ = CQ − BC = 11 cm − 7 cm = 4 cm

BR and BQ are tangents drawn from an external point B to the circle.

∴ BR = BQ

⇒ BR = 4 cm ( BQ = 4 cm)

#### Question 8:

In the given figure, Δ ABC is circumscribing a circle. Find the length of BC.

We are given the following figure

From the figure we get,

BC = BP + PC …… (1)

Now, let us find BP and PC separately.

From the property of tangents we know that when two tangents are drawn to a circle from a common external point, the length of the two tangents from the external point to the respective points of contact will be equal. Therefore we have,

BR = BP

It is given in the problem that BR = 3 cm. Therefore,

BP = 3 cm

Now let us find out PC.

Again using the same property of tangents which says that the length of two tangents drawn to a circle from the same external point will be equal, we have,

PC = QC…… (2)

From the figure we can see that,

QC = AC − AQ…… (3)

Again using the property that length of two tangents drawn to a circle from the same external point will be equal, we have,

AQ = AR

In the problem it is given that,

AR = 4 cm

Therefore,

AQ = 4 cm

Also, the length of AC is also given in the problem.

AC = 11 cm

Let us now substitute the values of AC and AQ in equation (3)

QC = 11 − 4

QC = 7

From equation (2) we can say that,

PC = 7

Finally, let us substitute the values of PC and BP in equation (1)

BC = BP + PC

BC = 3 + 7

BC = 10

Therefore, length of BC is 10 cm.

#### Question 9:

In the given figure, CP and CQ are tangents from an external point C to a circle with centre O. AB is another tangent which touches the circle at R. If CP = 11 cm and BR = 4 cm, find the length of BC.

[Hint: We have, CP = 11 cm

CP = CQ = CQ = 11 cm

Now, BR= BQ

BQ = 4 cm

BC = CQBQ = (11−4)cm = 7 cm

We are given the following figure

From the figure, we have

BC = CQ − BQ…… (1)

Let us now find out the values of CQ and BQ separately.

From the property of tangents we know that length of two tangents drawn from the same external point will be equal. Therefore,

CP = CQ

It is given in the problem that,

CP = 11

Therefore,

CQ = 11

Now let us find out the value of BQ.

Again from the same property of tangents, we know that length of two tangents drawn from the same external point will be equal. Therefore,

BR = BQ

It is given in the problem that,

BR = 4 cm

Therefore,

BQ = 4 cm

Now let us substitute the values of CQ and BQ in equation (1). We have,

BC = 11 − 4

BC = 7

Therefore, length of BC is 7 cm.

#### Question 10:

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Let us first draw whatever is given in the problem so that we can understand the problem better.

We have to find the length of AB, which is the chord of the larger circle which touches the smaller circle.

Clearly, OC is the radius of the smaller circle and is touching the tangent AB.

We know that the radius of the circle will always form a right angle with the tangent at the point of contact.

We have draw OA in order to complete the triangle OAC which will be a right triangle.

From the figure it is very clear that OA is the radius of the larger circle which is 5 cm.

We can now find AC using Pythagoras theorem. We have,

Similarly we can find CB. We have,

From the figure we can see that,

AB = AC + CB

Since we have found the values of AC and CB, let us substitute the values in the above equation. We get,

AB = 4 + 4

AB = 8

Therefore, the length of the chord of the larger circle which touches the smaller circle is 8 cm.

#### Question 11:

In the given figure, PA and PB are tangents to the circle with centre O such that $\angle \mathrm{APB}=50°$. Write the measure of $\angle \mathrm{OAB}.$           [CBSE 2015]

It is given that tangents PA and PB are drawn from an external point P to a circle with centre O.

∴ PA = PB          (Lengths of tangents drawn from an external point to a circle are equal)

In ∆PAB,

PA = PB

$\angle \mathrm{PBA}=\angle \mathrm{PAB}$     (In a triangle, equal sides have equal angles opposite to them)

Now,

$\angle \mathrm{PAB}+\angle \mathrm{PBA}+\angle \mathrm{APB}=180°$      (Angle sum property)

Now, PA is the tangent and OA is the radius through the point of contact A.

$\angle \mathrm{OAP}=90°$       (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

Now,

Hence, the measure of $\angle \mathrm{OAB}$ is 25º.

#### Question 12:

In Fig. 10.85, PQ is a chord of a circle and PT is the tangent at P such that $\angle$QPT = 600. Then , find  $\angle$PRQ .

figure

Construction: Take any point on major arc PQ and name it S. Join SQ and SP.
In the given figure, PT is the tangent. So, PT ⊥ PO.
$\angle \mathrm{QPT}=60°$
Thus, $\angle \mathrm{OPQ}=90°-60°=30°$
OQ = OP                    (Radii of the circle)
$\angle \mathrm{OQP}=\angle \mathrm{OPQ}=30°$

Now, $\angle \mathrm{PSQ}=\frac{1}{2}\angle \mathrm{POQ}=\frac{1}{2}×120=60°\phantom{\rule{0ex}{0ex}}$
PSQR is a cyclic quadrilateral. Thus,
$\angle \mathrm{PSQ}+\angle \mathrm{PRQ}=180°\phantom{\rule{0ex}{0ex}}⇒60°+\angle \mathrm{PRQ}=180°\phantom{\rule{0ex}{0ex}}⇒\angle \mathrm{PRQ}=120°$

#### Question 13:

In Fig. 10.86, PQL and PRM are tangents to the circle with centre O at the points Q and R respectively and S is a point on the circle such that   $\angle$SQL = 500 and  $\angle$SRM = 600. Then , find  $\angle$QSR.
figure

In the given figure,
PL and PM are the tangents to the circle with centre O.

Thus,
$\angle \mathrm{ORS}=90°-60°=30°\phantom{\rule{0ex}{0ex}}\angle \mathrm{OQS}=90°-50°=40°$
OQ = OS
So, $\angle \mathrm{OQS}=\angle \mathrm{OSQ}=40°$
Similarly.
OS = OR
So, $\angle \mathrm{ORS}=\angle \mathrm{OSR}=30°$
$\angle \mathrm{RSQ}=\angle \mathrm{QSO}+\angle \mathrm{RSO}\phantom{\rule{0ex}{0ex}}⇒\angle \mathrm{RSQ}=40°+30°=70°$

#### Question 14:

In Fig. 10.87 , BOA is a diameter of  a circle and the tangent at a point P meets BA produced at T . If $\angle$PBO = 300 , then find  $\angle$PTA .

figure

In the given figure, PT is the tangent to the circle with centre O.
$\angle$PBO = 30$°$
So, $\angle$PBO = $\angle$BPO = 30$°$
$\angle$BPA = 90$°$            (Angle made by the diameter on the arc of the circle)
In ∆APB,
$\angle$BPA + $\angle$PBO + $\angle$PAB = 180$°$
⇒ 90$°$ + 30$°$$\angle$PAB = 180$°$
⇒ $\angle$PAB = 60$°$
In ∆OPA,
So, $\angle$OPA = OAP = 60$°$
Hence, $\angle$AOP = 180$°$ − 60$°$ − 60$°$ = 60$°$
$\angle$OPT + $\angle$PTO + $\angle$POT = 180$°$
⇒ 90$°$ + $\angle$PTO + 60$°$ = 180$°$
⇒ $\angle$PTO = 30$°$