Question

In the given figure, the area of the shaded portion is (π – 1)R², where R is the radius of the circle with centre at O. What is one of the acute angles of ΔABC ? 

(a) 30º 

(b) 45º 

(c) 60º 

(d) Cannot be determined.

Answer 1

(b) 45°

Area of triangle ABC 

= Area of circle – Area of shaded region 

= πR² – (π – 1) R² = R²

But Area of ΔABC = \(\frac{1}{2}\) x AB x AC

(∠ABC = 90°, angle in a semicircle is a right angle)

⇒   \(\frac{1}{2}\) x AB x AC = R² ⇒ AB x BC = 2R²

In ΔABC, AC² = AB² + BC² 

⇒ (2R)² = AB² + BC² 

⇒ 2 × AB × BC = AB² + BC²           [Using (i)] 

⇒ AB² + BC² – 2AB BC = 0 

⇒ (AB – BC)² = 0 

⇒ AB = BC = √2 R

∴ tan A = \(\frac{BC}{AB}\) = \(\frac{\sqrt2R}{\sqrt2R}\) = 1 ⇒ ∠A = 45°.