An isosceles triangle with base 6 cms and base angles 30 degree each is incribed in a circle - Maths - Inverse Trigonometric Functions

Question

An isosceles triangle with base 6 cms and base angles 30 degree
each is incribed in a circle A second circle, which is situated
outside the triangle, touches the first circle and also touches the
base of the triangle at its midpoint Find its radius

Aakash Sharma · Accepted Answer

Let ABC be the isosceles triangle with AB = AC and BC = 6 cm

also ∠ABC = ∠ACB = 30°

and let the Radius of the first and second circle be R and r respectively.

Let D be the point of contact of second circle.

$\Rightarrow BD = DC = \frac{BC}{2} = 3 cm$

In ∆ABD and ∆ACB

AB = AC

∠ABD = ∠ACB = 30°

BD = DC = 3 cm

Thus ∆ABD ≅ ∆ACB (by SAS congruence criterion)

⇒ ∠ADB = ∠ADC

But ∠ADB + ∠ADC = 180° (∵ BC is a straight line)

⇒ ∠ADB = ∠ADC = $\frac{180 °}{2} = 90 °$

Hence AD is the perpendicular bisector of BC and will pass through the centre of first circle.

⇒ AE is the diameter of first circle

⇒ AE = 2 R

Also ∠ADB + ∠BDE = 180° (Linear pair)

⇒ 90° + ∠BDE = 180°

⇒ ∠BDE = 180° – 90° = 90°

Hence BC is tangent to second circle and DE is the diameter of second circle.

⇒ DE = 2 r

Now in ∆ ABD

$\cos 30 ° = \frac{BD}{AB} \Rightarrow \frac{\sqrt{3}}{2} = \frac{3 cm}{AB} \Rightarrow AB = 2 \sqrt{3} cm a nd tan 30°= \frac{AD}{BD} \Rightarrow \frac{1}{\sqrt{3}} = \frac{AD}{3} \Rightarrow AD = \sqrt{3}$

Also ∠ABD + ∠ADB + ∠BAD = 180° (Angle sum property)

⇒ 30° + 90° + ∠BAD = 180°

⇒ ∠BAD = 180° – 120° = 60°

Now in ∆ABE

$\cos 60 ° = \frac{AB}{AE} \Rightarrow \frac{1}{2} = \frac{2 \sqrt{3}}{2 R} \Rightarrow R = 2 \sqrt{3} cm$

Now AE = AD + DE

$\Rightarrow 2 R = \sqrt{3} + 2 r \Rightarrow 2 r = 4 \sqrt{3} - \sqrt{3} = 3 \sqrt{3} \Rightarrow r = \frac{3 \sqrt{3}}{2} cm$

Hence the radius of first circle is $2 \sqrt{3}$ cm and second circle is $\frac{3 \sqrt{3}}{2}$ cm.

1

An isosceles triangle with base 6 cms. and base angles 30 degree each is incribed in a circle. A second circle, which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.