SATHEE: Properties of Triangles 2 Question 3

Properties of Triangles 2 Question 3

3. In radius of a circle which is inscribed in a isosceles triangle one of whose angle is $2 π / 3$ , is $\sqrt{3}$ , then area of triangle (in sq units) is

(2006, 2M)

(a) $4 \sqrt{3}$

(b) $12 - 7 \sqrt{3}$

(d) None of these

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Answer:

Correct Answer: 3. (c)

Solution:

Let $A B = A C = a$ and $∠ A = 120^{\circ}$ .

$∴$ Area of triangle $= \frac{1}{2} a^{2} \sin 120^{\circ}$

where, $a = A D + B D = \sqrt{3} \tan 30^{\circ} + \sqrt{3} \cot 15^{\circ}$

$= 1 + \frac{\sqrt{3}}{\tan (45^{\circ} - 15^{\circ})}$

$\begin{aligned} \Rightarrow a & = 1 + \sqrt{3} \frac{1 + \tan 45^{\circ} \tan 30^{\circ}}{\tan 45^{\circ} - \tan 30^{\circ}} \\ = 1 + \sqrt{3} \frac{\sqrt{3} + 1}{\sqrt{3} - 1} = 4 + 2 \sqrt{3} \end{aligned}$

$∴$ Area of a triangle

$= \frac{1}{2} (4 + 2 \sqrt{3})^{2} \frac{\sqrt{3}}{2} = (12 + 7 \sqrt{3}) sq units$

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Properties of Triangles 2 Question 3

3. In radius of a circle which is inscribed in a isosceles triangle one of whose angle is 2π/3, is 3, then area of triangle (in sq units) is

Answer:

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3. In radius of a circle which is inscribed in a isosceles triangle one of whose angle is $2 π / 3$ , is $\sqrt{3}$ , then area of triangle (in sq units) is