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#### Question 18:

Construct a ΔABC in which AB = 5 cm. ∠B = 60° altitude CD = 3cm. Construct a ΔAQR similar to ΔABC such that side ΔAQR is 1.5 times that of the corresponding sides of ΔACB.

Given that

Construct a trianglein which and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With as centre and draw an angle.

Step: III -From point A and B construct  which cut the line BS at point C

Step: IV- Join AC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off five points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle  each of whose sides is of the corresponding sides of.

So, we draw a line  on AX from point which is  and meeting AB at Q.

Step: IX- From point draw  and meeting AC at R

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 1:

Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.

Given that

Construct a circle of radius, and form its centre, construct the pair of tangents to the circle.

Find the length of tangents.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a circle of radius.

Step: II- Make a point P at a distance of, and join .

Step: III -Draw a right bisector of, intersecting at Q .

Step: IV- Taking Q as centre and radius, draw a circle to intersect the given circle at T and T’.

Step: V- Joins PT and PT’ to obtain the require tangents.

Thus, are the required tangents.

Find the length of tangents.

As we know that and is right triangle.

Therefore,

In,

Thus, the length of tangents

#### Question 2:

Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.

Given that

Construct a circle of radius, and extended diameter each at distance of 7cm from its centre. Construct the pair of tangents to the circle from these two points .

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a circle of radius.

Step: II- Make a line CD .

Step: III-Extend the line CD in such a way that point

Step: IV- CP at a distance of, and join draw a right bisector of, intersecting at R.

Step V:- Similarly, DQ at a distance of, and join draw a right bisector of, intersecting at S.

Step VI: Taking R and S as centre and radius, draw a circle to intersect the given circle at T and T’

B and B ’respectively.

Step: VII- Joins PT and PT’ as well as QB and QB’ to obtain the require tangents.

Thus, are the required tangents.

#### Question 3:

Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

Given that

Construct a circle of radius, and extended diameter each at distance of 7cm from its centre. Construct the pair of tangents to the circle from these two points .

We follow the following steps to construct the given

Step of construction

Step: I First of all we draw a line.

Step: II taking A as a centre and draw a circle of radius. Similarly, taking B as a centre and draw a circle of radius.

Step: III draw the perpendicular bisector of

Step IV : draw the another circle with taking the bisector point as centre and radius = mid point of which cut the point

Step: V joins and respectively. as well as to obtain the require tangents.

Thus, are the required tangents.

#### Question 4:

Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 6.2 from its centre.                          [CBSE 2013]

Steps of Construction

Step 1. Draw a circle with O as centre and radius 3.5 cm.

Step 2. Mark a point P outside the circle such that OP = 6.2 cm.

Step 3. Join OP. Draw the perpendicular bisector XY of OP, cutting OP at Q.

Step 4. Draw a circle with Q as centre and radius PQ (or OQ), to intersect the given circle at the points T and T'.

Step 5. Join PT and PT'.

Here, PT and PT' are the required tangents.

#### Question 5:

Draw a pair of tangents to a circle of radius 4.5 cm, which are inclined to each other at an angle of 45°.                         [CBSE 2013]

Steps of Construction

Step 1. Draw a circle with centre O and radius 4.5 cm.

Step 2. Draw any diameter AOB of the circle.

Step 3. Construct $\angle$BOC = 45º such that radius OC cuts the circle at C.

Step 4. Draw AM ⊥ AB and CN ⊥ OC. Suppose AM and CN intersect each other at P.

Here, AP and CP are the pair of tangents to the circle inclined to each other at an angle of 45º.

#### Question 6:

Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and $\angle$B = 90º. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle.                                                                                         [CBSE 2014]

Steps of Construction

Step 1. Draw a line segment AB = 6 cm.

Step 2. At B, draw $\angle$ABX = 90º.

Step 3. With B as centre and radius 8 cm, draw an arc cutting ray BX at C.

Step 4. Join AC. Thus, ∆ABC is the required triangle.

Step 5. From B, draw BD ⊥ AC.

Step 6. Draw the perpendicular bisector of BC, cutting BC at O.

Step 7. With O as centre and radius OB (or OC), draw a circle. This circle passes through B, C and D.

Thus, this is the required circle.

Step 8. Join OA.

Step 9. Draw the perpendicular bisector of OA, cutting OA at E.

Step 10. With E as centre and radius AE (or OE), draw a circle intersecting the circle with centre O at B and F.

Step 11. Join AF.

Here, AB and AF  are the required tangents.

#### Question 7:

Draw two concentric circles of radii 3 cm and 5 cm. Construct a tangent to smaller circle from a point on the larger circle. Also measure its length.

Following are the steps to draw tangents on the given circle:

Step 1

Draw a circle of 3 cm radius with centre O on the given plane.

Step 2

Draw a circle of 5 cm radius, taking O as its centre. Locate a point P on this circle and join OP.

Step 3

Bisect OP. Let M be the midpoint of PO.

Step 4

Taking M as its centre and MO as its radius, draw a circle. Let it intersect the given circle at points Q and R.

Step 5

Join PQ and PR. PQ and PR are the required tangents.

It can be observed that PQ and PR are of length 4 cm each.

In ΔPQO,

Since PQ is a tangent,

∠PQO = 90°

PO = 5 cm

QO = 3 cm

Applying Pythagoras theorem in ΔPQO, we obtain

PQ2 + QO2 = PQ2

PQ2 + (3)2 = (5)2

PQ+ 9 = 25

PQ= 25 − 9

PQ= 16

PQ = 4 cm

Hence justified.

#### Question 1:

To draw a pair of tangents to a circle which are inclined to each other at an angle of 60°, it is required to draw tangents at end points of those two radii of the circle, the angle between them should be
(a) 135°
(b) 90°
(c) 60°
(d) 120°

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

∠APB = 60°

Now,

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

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

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

⇒ ∠AOB + 90º + 90º + 60º = 360º

⇒ ∠AOB = 360º − 240º = 120º

Thus, in order to draw a pair of tangents to a circle which are inclined to each other at an angle of 60°, it is required to draw tangents at end points of those two radii of the circle, the angle between them should be 120°.

Hence, the correct answer is option (d).

#### Question 2:

To divide a line segment AB in the ratio 5:7, first ray AX is drawn so that ∠BAX is an acute angle and then at equal distances points are marked on the ray AX such that the minimum number of these points is
(a) 8
(b) 10
(c) 11
(d) 12

We know that, in order to divide a line segment AB in the ratio m : n (m, n are positive integers), first draw a ray AX such that ∠BAX is an acute angle and then mark points on ray AX at equal distances such that the minimum number of these points is m + n.

Here, m = 5 and n = 7

m + n = 5 + 7 = 12

Thus, to divide a line segment AB in the ratio 5 : 7, first ray AX is drawn so that ∠BAX is an acute angle and then at equal distances points are marked on the ray AX such that the minimum number of these points is 12.

Hence, the correct answer is option (d).

#### Question 3:

To divide a line segment AB in the ratio m : n (m, n are positive integers), draw a ray AX such that ∠BAX is an acute angle and then mark points on ray AX at equal distances such that the minimum number of these points is
(a) greater than m and n
(b) m + n
(c) m + n – 1
(d) mn

To divide a line segment AB in the ratio m : n (m, n are positive integers), firstly draw a ray AX such that ∠BAX is an acute angle. Then, along AX mark off m + n points at equal distances.

Here, AP : PB = m : n

Thus, to divide a line segment AB in the ratio m : n (m, n are positive integers), draw a ray AX such that ∠BAX is an acute angle and then mark points on ray AX at equal distances such that the minimum number of these points is m + n.

Hence, the correct answer is option (b).

#### Question 4:

To draw a pair of tangents to a circle which are inclined to each other at an angle of 35°, it is required to draw tangents at the end points of those two radii of the circle, the angle between which is
(a) 105°
(b) 70°
(c) 140°
(d) 145°

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

∠APB = 35°

Now,

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

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

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

⇒ ∠AOB + 90º + 90º + 35º = 360º

⇒ ∠AOB = 360º − 215º = 145º

Thus, in order to draw a pair of tangents to a circle which are inclined to each other at an angle of 35°, it is required to draw tangents at the end points of those two radii of the circle, the angle between which is 145º.

Hence, the correct answer is option (d).

#### Question 5:

To construct a triangle similar to a given ∆ABC with its side 8/5 of the corresponding side of ∆ABC draw a ray BX such that ∠CBX is an acute angle and X is on the opposite side of A with respect to BC. The minimum number of points to be located at equal distances on ray BX is
(a) 5
(b) 8
(c) 13
(d) 3

In order to construct a triangle similar to a given triangle with its sides $\frac{m}{n}$ of the corresponding sides of the triangle, the minimum number number of points to be located at equal distances on the ray is m or n, whichever is greater.

If m > n, then minimum points to be located at equal distances on the ray is m.

If n > m, then minimum points to be located at equal distances on the ray is n.

Here, m = 8 and n = 5

8 > 5

Thus, in order to construct a triangle similar to a given ∆ABC with its side $\frac{8}{5}$ of the corresponding side of ∆ABC, draw a ray BX such that ∠CBX is an acute angle and X is on the opposite side of A with respect to BC. The minimum number of points to be located at equal distances on ray BX is 8.

Hence, the correct answer is option (b).

#### Question 6:

To divide a line segment AB in the ratio 4 : 7, a ray AX is drawn first such that ∠BAX is an acute angle and then points A1, A2, A3, ... are located at equal distances on the ray AX and the point B is joined to
(a) A12
(b) A11
(c) A10
(d) A9

To divide a line segment AB in the ratio m : n (m, n are positive integers), firstly draw a ray AX such that ∠BAX is an acute angle. Now, along AX mark off m + n points at equal distances.
Suppose these m + n points marked on ray AX be A1, A2, A3, ..., Am, Am + 1, ..., Am + n. Then, join B to Am + n.

Here, m = 4 and n = 7

m + n = 4 + 7 = 11

So, B is joined to the point A11.

Thus, in order to divide a line segment AB in the ratio 4 : 7, a ray AX is drawn first such that ∠BAX is an acute angle and then points A1, A2, A3, ... are located at equal distances on the ray AX and the point B is joined to A11.

Hence, the correct answer is option (b).

#### Question 7:

To construct a triangle similar to a given ∆ABC with its sides 3/7 of the corresponding sides of ∆ABC, first draw a ray BX such that ∠CBX is an acute angle and X lies on the opposite side of A with respect to BC. Then locate points B1, B2, B3,.... on BX at equal distances and next step is to join
(a) B10 to C
(b) B3 to C
(c) B7 to C
(d) B4 to C

In order to construct a triangle similar to a given triangle with its sides $\frac{m}{n}$ of the corresponding sides of the triangle, the minimum number number of points to be located at equal distances on the ray is m or n, whichever is greater.

If m > n, then minimum points to be located at equal distances on the ray is m.

If n > m, then minimum points to be located at equal distances on the ray is n.

Here, m = 3 and n = 7

7 > 3

So, seven points B1, B2, B3, B4, B5, B6, B7 are marked at equal distance on BX. Then B7 is joined to C.

Thus, in order to construct a triangle similar to a given ∆ABC with its sides $\frac{3}{7}$ of the corresponding sides of ∆ABC, first draw a ray BX such that ∠CBX is an acute angle and X lies on the opposite side of A with respect to BC. Then locate points B1, B2, B3,...., B7 on BX at equal distances and next step is to join B7 to C.

Hence, the correct answer is option (c).

#### Question 8:

To divide a line segment AB in the ratio 5 : 6, draw a ray AX such that ∠BAX is an acute angle, then draw a ray BY parallel to AX and the points A1, A2, .... and B1, B2,..... are located at equal distances on rays AX and BY respectively. Then the points joined are
(a) A5 and B6
(b) A6 and B5
(c) A4 and B5
(d) A5 and B4

To divide a line segment AB in the ratio m : n, draw a ray AX such that ∠BAX is an acute angle. Then draw a ray BY parallel to AX. The points A1, A2, ..., Am and B1, B2, ..., Bn are located at equal distances on rays AX and BY respectively. Then the points Am and Bn are joined to divide the line segment AB in the ratio m : n.

Here, m = 5 and n = 6

So, the points A1, A2, ..., A and B1, B2, ..., B6 are located at equal distances on rays AX and BY respectively. Then the points A5 and B6 are joined.

Thus, in order to divide a line segment AB in the ratio 5 : 6, draw a ray AX such that ∠BAX is an acute angle, then draw a ray BY parallel to AX and the points A1, A2, ..., A5 and B1, B2, ..., B6 are located at equal distances on rays AX and BY respectively. Then the points joined are A5 and B6.

Hence, the correct answer is option (a).

#### Question 1:

Determine a point which divides a line segment of length 12 cm internally in the ratio 2 : 3 Also, justify your construction.

Given that

Determine a point which divides a line segment of lengthinternally in the ratio of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- We draw a ray making an acute anglewith.

Step: III- Draw a ray parallel to AX by making an acute angle.

Step IV- Mark of two points on and three points on in such a way that.

Step: V- Joins and this line intersects at a point P.

Thus, P is the point dividing internally in the ratio of

Justification:

In we have

And

So, AA similarity criterion, we have

#### Question 2:

Divide a line segment of length 9 cm internally in the ratio 4 : 3. Also, give justification of the construction.

Given that

Determine a point which divides a line segment of lengthinternally in the ratio of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- We draw a ray making an acute anglewith.

Step: III- Draw a ray parallel to AX by making an acute angle.

Step IV- Mark of two points on and three points on in such a way that.

Step: V- Joins and this line intersects at a point P.

Thus, P is the point dividing internally in the ratio of

Justification:

In we have

And

So, AA similarity criterion, we have

#### Question 3:

Divide a line segment of length 14 cm internally in the ratio 2 : 5. Also, justify your construction.

Given that

Determine a point which divides a line segment of lengthinternally in the ratio of.

We follow the following steps to construct the given

Step of construction

Step: I-First of all we draw a line segment.

Step: II- We draw a ray making an acute anglewith.

Step: III- Draw a ray parallel to AX by making an acute angle.

Step IV- Mark of two points on and three points on in such a way that.

Step: V- Joins and this line intersects at a point P.

Thus, P is the point dividing internally in the ratio of

Justification:

In we have

And

So, AA similarity criterion, we have

#### Question 4:

Draw a line segment of length 8 cm and divide it internally in the ratio 4:5.

Steps of construction:

1) Draw a line segment AB = 8 cm.

2) Draw a ray AX making an acute angle $\angle BAX=60°$ with AB.

3) Draw a ray BY parallel to AX by making an acute angle $\angle ABY=\angle BAX$.

4) Mark of four points on AX and five points on BY in such a way that $A{A}_{1}={A}_{1}{A}_{2}={A}_{2}{A}_{3}={A}_{3}{A}_{4}$.

5) Joins ${A}_{4}{B}_{5}$ and this line intersects AB at a point P.

Thus, P is the point dividing AB internally in the ratio of 4 : 5.

#### Question 1:

Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are (2/3)  of the corresponding sides of it.

Given that

Construct a triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of it.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and radius, draw an arc.

Step: III- With B as centre and radius, draw an arc, intersecting the arc drawn in step II at C.

Step: IV- Joins AC and BC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off three points such that

Step: VII- Join.

Step: VIII- Since we have to construct a triangle each of whose sides is two-third of the corresponding sides of.

So, we take two parts out of three equal parts on AX from point draw and meeting AB at C’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is two third of the corresponding sides of.

#### Question 2:

Construct a triangle similar to a given ΔABC such that each of its sides is (5/7)th  of the corresponding sides of Δ ABC. It is given that AB - 5 cm, BC = 7 cm and ∠ABC = 50°.

Given that

Construct a triangle similar to a triangle ABC such that each of sides is of the corresponding sides of triangle ABC.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With B as centre and draw an angle.

Step: III- With B as centre and radius, draw an arc, cut the line BY drawn in step II at C.

Step: IV- Joins AC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off seven points such that

Step: VII-Join.

Step: VIII- Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take five parts out of seven equal parts on AX from point draw and meeting AB at B’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 3:

Construct a triangle similar to a given ΔABC such that each of its sides is (2/3)rd of the corresponding sides of ΔABC. It is given that BC = 6 cm, ∠B = 50° and ∠C = 60°.

Given that

Construct a triangle of given data, and then a triangle similar to it whose sides are of the corresponding sides of .

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With B as centre draw an angle.

Step: III- With C as centre draw an anglewhich intersecting the line drawn in step II at A.

Step: IV- Joins AB and AC to obtain.

Step: V -Below BC, makes an acute angle.

Step: VI -Along BX, mark off three points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle each of whose sides is two-third of the corresponding sides of.

So, we take two parts out of three equal parts on BX from point draw and meeting BC at C’.

Step: IX -From C’ draw and meeting AB at A

Thus, is the required triangle, each of whose sides is two third of the corresponding sides of.

#### Question 4:

Draw a ΔABC in which BC = 6 cm, AB = 4 cm and AC = 5 cm. Draw a triangle similar to ΔABC with its sides equal to (3/4)th of the corresponding sides of ΔABC.

Given that

Construct a triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and radius, draw an arc.

Step: III -With B as centre and radius, draw an arc, intersecting the arc drawn in step II at C.

Step: IV -Joins AC and BC to obtain.

Step: V -Below AB, makes an acute angle.

Step: VI -Along AX, mark off four points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take three parts out of four equal parts on AX from point draw and meeting AB at B’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 5:

Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are $\frac{5}{7}$ of the corresponding sides of the first triangle.

Steps of Construction

Step 1.
Draw a line segment QR = 6 cm.

Step 2. With Q as centre and radius 7 cm, draw an arc.

Step 3. With R as centre and radius 5 cm, draw an arc cutting the previous arc at P.

Step 4. Join PQ and PR. Thus ∆PQR is the required triangle.

Step 5. Below QR, draw an acute angle ∠RQX.

Step 6. Along QX, mark seven points Q1, Q2, Q3, Q4, Q5, Q6 and Q7 such that QQ1 = Q1Q2 = Q2Q3 = Q3Q4 = Q4Q5 = Q5Q6 = Q6Q7.

Step 7. Join Q7R.

Step 8. From Q5, draw Q5R' || Q7R meeting QR at R'.

Step 9. From R', draw P'R' || PR meeting PQ in P'.

Here, ∆P'QR' is the required triangle, each of whose sides are $\frac{5}{7}$ of the corresponding sides of ∆PQR.

#### Question 6:

Draw a right triangle ABC in which AC = AB = 4.5 cm and ∠A = 90°. Draw a triangle similar to ΔABC with its sides equal to (5/4)th of the corresponding sides of ΔABC.

Given that

Construct a right triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and draw an angle.

Step: III- With A as centre and radius.

Step: IV- Join BC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off five points such that

Step: VII-Join.

Step: VIII- Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we draw a line on AX from point which is and meeting AB at B’.

Step: IX- From B’ point draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 7:

Draw a right triangle in which the sides (other than hypotenuse) are of lengths 5 cm and 4 cm. Then construct another triangle whose sides are 5/3 times the corresponding sides of the given triangle.

Given that

Construct a right triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and draw an angle.

Step: III- With A as centre and radius.

Step: IV -Join BC to obtain.

Step: V -Below AB, makes an acute angle.

Step: VI -Along AX, mark off five points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we draw a line on AX from point which is and meeting AB at B’.

Step: IX -From B’ point draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 8:

Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are 3/2 times the corresponding sides of the isosceles triangle.

Given that

Construct an isosceles triangle ABC in which AB = BC = 6 cm and altitude = 4 cm then another triangle similar to it whose sides are $\frac{3}{2}$ of the corresponding sides of$△ABC$.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment AB = 6 cm.

Step: II- With B as centre and radius = BC = 6 cm, draw an arc.

Step: III- From point A and B construct which cut the line BS at point C

Step: IV -Join AC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI -Along AX, mark off five points such that

Step: VII- Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we draw a line on AX from point which is and meeting AB at Q.

Step: IX -From Q point draw and meeting AC at R

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 9:

Draw a ΔABC with side BC = 6 cm. AB = 5 cm and ∠ ABC = 60°. Then, construct a triangle whose sides are (3/4)th of the corresponding sides of the ΔABC.

Given that

Construct a of given data, and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With B as centre draw an angle.

Step: III- With B as centre and radius, draw an arc.

Step: IV- Join AC to obtain.

Step: V -Below AB, makes an acute angle.

Step: VI -Along AX, mark off four points such that

Step: VII -Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take three parts out of four equal parts on AX from point draw and meeting AB at B’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 10:

Construct a triangle similar to Δ ABC in which AB = 4.6 cm, BC = 5.1 cm, ∠A = 60° with scale factor 4 : 5.

Given that

Construct a of given data, and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre draw an angle.

Step: III- With B as centre and radius, draw an arc, intersecting the arc drawn in step II at C.

Step: IV- Joins BC to obtain.

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off five points such that

Step: VII- Join.

Step: VIII- Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take four parts out of five equal parts on AX from point draw and meeting AB at B’.

Step: IX- From B’ draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 11:

Construct a triangle similar to a given ΔXYZ with its sides equal to (3/4)th of the corresponding sides of ΔXYZ. Write the steps of construction.

Given that

Construct a of given data, and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With Y as centre draw an angle.

Step: III- With Y as centre and radius, draw an arc.

Step: IV- Join XZ to obtain.

Step: V- Below XY, makes an acute angle.

Step: VI -Along XP, mark off four points such that

Step: VII- Join.

Step: VIII- Since we have to construct a triangle each of whose sides is of the corresponding sides of.

So, we take three parts out of four equal parts on XP from point draw and meeting XY at Y’.

Step: IX- From Y’ draw and meeting XZ at Z

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 12:

Draw a right triangle in which sides (other than the hypotenuse) are of lengths 8 cm and 6 cm. Then construct another triangle whose sides are $\frac{3}{4}$ times the corresponding sides of the first triangle.

Given that

Construct a right triangle of sides and then a triangle similar to it whose sides are of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment.

Step: II- With A as centre and draw an angle.

Step: III- With A as centre and radius.

Step: IV-Join BC to obtain right .

Step: V- Below AB, makes an acute angle.

Step: VI- Along AX, mark off five points such that

Step: VII- Join.

Step: VIII -Since we have to construct a triangle each of whose sides is of the corresponding sides of right .

So, we draw a line on AX from point which is and meeting AB at B’.

Step: IX- From B’ point draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is of the corresponding sides of.

#### Question 13:

Construct a triangle with sides 5 cm, 5.5 cm and 6.5 cm. Now construct another triangle, whose sides are $\frac{3}{5}$ times the corresponding sides of the given triangle.                                                                                                                                                                            [CBSE 2014]

Steps of Construction

Step 1. Draw a line segment QR = 5.5 cm.

Step 2. With Q as centre and radius 5 cm, draw an arc.

Step 3. With R as centre and radius 6.5 cm, draw an arc cutting the previous arc at P.

Step 4. Join PQ and PR. Thus, ∆PQR is the required triangle.

Step 5. Below QR, draw an acute angle $\angle$RQX.

Step 6. Along QX, mark five points R1, R2, R3, R4 and R5 such that QR1 = R1R2 = R2R3 = R3R4 = R4R5.

Step 7. Join RR5.

Step 8. From R3, draw R3R' || RR5 meeting QR at R'.

Step 9. From R', draw P'R' || PR meeting PQ in P'.

Here, ∆P'QR' is the required triangle, each of whose sides are $\frac{3}{5}$ times the corresponding sides of ∆PQR.

#### Question 14:

Construct a triangle PQR with sides QR = 7 cm, PQ = 6 cm and $\angle$PQR = 60º. Then construct another triangle whose sides are $\frac{3}{5}$ of the corresponiding sides of ∆PQR.                                                                                                                                         [CBSE 2014, 2015]

Steps of Construction

Step 1. Draw a line segment QR = 7 cm.

Step 2. At B, draw $\angle$XQR = 60º.

Step 3. With Q as centre and radius 6 cm, draw an arc cutting the ray QX at P.

Step 4. Join PR. Thus, ∆PQR is the required triangle.

Step 5. Below QR, draw an acute angle $\angle$YQR.

Step 6. Along QY, mark five points R1, R2, R3, R4 and R5 such that QR1 = R1R2 = R2R3 = R3R4 = R4R5 .

Step 7. Join RR5.

Step 8. From R3, draw R3R' || RR5 meeting QR at R'.

Step 9. From R', draw P'R' || PR meeting PQ in P'.

Here, ∆P'QR' is the required triangle whose sides are $\frac{3}{5}$ of the corresponding sides of ∆PQR.

#### Question 15:

Draw a ∆ABC in which base BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct another triangle whose sides are $\frac{3}{4}$ of the corresponding sides of ∆ABC.

ΔA'BC' whose sides are of the corresponding sides of ΔABC can be drawn as follows.

Step 1

Draw a ΔABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°.

Step 2

Draw a ray BX making an acute angle with BC on the opposite side of vertex A.

Step 3

Locate 4 points (as 4 is greater in 3 and 4), B1, B2, B3, B4, on line segment BX.

Step 4

Join B4C and draw a line through B3, parallel to B4C intersecting BC at C'.

Step 5

Draw a line through C' parallel to AC intersecting AB at A'. ΔA'BC' is the required triangle.

Justification

The construction can be justified by proving

In ΔA'BC' and ΔABC,

∠A'C'B = ∠ACB (Corresponding angles)

∠A'BC' = ∠ABC (Common)

∴ ΔA'BC' ∼ ΔABC (AA similarity criterion)

… (1)

In ΔBB3C' and ΔBB4C,

∠B3BC' = ∠B4BC (Common)

∠BB3C' = ∠BB4C (Corresponding angles)

∴ ΔBB3C' ∼ ΔBB4C (AA similarity criterion)

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

#### Question 16:

Draw a right triangle in which the sides (other than the hypotenuse) are of lengths 4 cm and 3 cm. Now construct another triangle whose sides are $\frac{3}{5}$ times the corresponding sides of the given triangle.

Construct a right triangle of sides  and then a triangle similar to it whose sides are ${\left(\frac{3}{5}\right)}^{\mathrm{th}}$of the corresponding sides of.

We follow the following steps to construct the given

Step of construction

Step: I- First of all we draw a line segment AB = 4 cm.

Step: II- With A as centre and draw an angle.

Step: III- With A as centre and radius AC = 3 cm.

Step: IV-Join BC to obtain right .

Step: V- Below AB, makes an acute angle $\angle BAX$.

Step: VI- Along AX, mark off five points  such that $={A}_{4}{A}_{5}$.

Step: VII- Join ${A}_{5}B$.

Step: VIII -Since we have to construct a triangle each of whose sides is ${\left(\frac{3}{5}\right)}^{\mathrm{th}}$ of the corresponding sides of right .

So, we draw a line on AX from point which is ${A}_{3}B\parallel {A}_{5}B$  and meeting AB at B’.

Step: IX- From B’ point draw and meeting AC at C’

Thus, is the required triangle, each of whose sides is ${\left(\frac{3}{5}\right)}^{\mathrm{th}}$ of the corresponding sides of.

#### Question 17:

Construct an equilateral triangle with each side 5 cm. Then construct another triangle whose sides are 2/3 times the corresponding sides of ∆ABC.

Steps of Construction

Step 1.
Draw a line segment BC = 5 cm.

Step 2. With B as centre and radius 5 cm, draw an arc.

Step 3. With C as centre and radius 5 cm, draw an arc cutting the previous arc at A.

Step 4. Join AB and AC. Thus, ∆ABC is the required equilateral triangle.

Step 5. Below BC, draw an acute angle ∠CBX.

Step 6. Along BX, mark three points B1, B2 and B3 such that BB1 = B1B2 = B2B3.

Step 7. Join B3C.

Step 8. From B2, draw B2C' || B3C meeting BC at C'.

Step 9. From C', draw A'C' || AC meeting AB in A'.

Here, ∆A'BC' is the required triangle, each of whose sides are $\frac{2}{3}$ times the corresponding sides of ∆ABC.

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