Question

In the figure given alongside, ΔABC is equilateral with area S. M is the mid-point of BC and P is a point on AM extended such that MP = BM. If the semi-circle on AP intersects CB extended at Q and the area of a square with MQ as a side is T, which of the following is true? 

(a) T = √2S

(b) T = S 

(c) T = √3S

(d) T = 2S

Answer 1

Answer : (b) T = S 

∠PQA = 90º (Angle in a semi-circle) 

∴ (MP) . (MA) = MQ². (Mean Proportional) ...(i) 

Let MP = MB = 1(Given MP = BM) 

Since ΔABC is an equilateral Δ, 

AM = \(\frac{\sqrt{3}}{2}\) × BC = \( \sqrt{3}\) . 

∴ From (i) 1 . \( \sqrt{3}\) = MQ² → MQ =  \(3^{\frac{1}{4}}\)

Area of square = T = (MQ)² = \( \sqrt{3}\) 

Area of ΔABC = S = \(\frac{\sqrt{3}}{4}\) (BC)² 

=  \(\frac{\sqrt{3}}{4}\) × 4 = \( \sqrt{3}\)  

∴ T = S.