Given an isosceles triangle, whose one angle is 120° and radius of its incircle $=\sqrt{3}$. Then the area of the triangle in sq. units is

JEE Advanced · 2006 · 09 Apr 2006 · Properties of Triangle

Given an isosceles triangle, whose one angle is 120° and radius of its incircle

= 3

. Then the area of the triangle in sq. units is

(A) $7 + 123$
(B) $12 - 73$
(C) $12 + 73$ ✓ Correct
(D) $4 π$

Answer: (C)

Solution

Let the isosceles triangle have sides

a, b, b

with the angle between equal sides being 120°.

Using the sine rule:

\frac{s i n 120°}{a} = \frac{s i n 30°}{b}

, which gives

a = 3 b

.

The area is

Δ = \frac{1}{2} b^{2} sin 120° = \frac{3}{4} b^{2}

.

The semi-perimeter is

s = \frac{a + 2 b}{2} = \frac{3 b + 2 b}{2}

.

Using

Δ = r s

where

r = 3

\frac{3}{4} b^{2} = 3 \cdot \frac{( 3 + 2 ) b}{2}

Solving for

b

and substituting back gives

Δ = 12 + 73

JEE Advanced · 2006 · 09 Apr 2006 · Single correct option

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