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In the figure given below, ABC is a right-angled triangle where ∠ A = 90°, AB = p cm and AC = q cm. On the three sides as diameters semicircles are drawn as shown in the figure. The area of the shaded portion, in square cm, is

In the figure given below, ABC is a right-angled triangle where ∠ A = 90°, AB = p cm and AC = q cm. On the three sides as diameters semicircles are drawn as shown in the figure. The area of the shaded portion, in square cm, is Correct Answer pq/2

Given, AB = p cm and AC = q cm

∵ Area of right-angled triangle = 1/2 × Base × Height

⇒ Area of triangle ABC = pq/2

Applying Pythagoras theorem in ∆ABC,

⇒ BC2 = AB2 + AC2

⇒ BC2 = (p2 + q2)

∵ Area of semi-circle = (π/8) × (diameter)2

⇒ Area of semi-circle with diameter AB = (π/8) × (p)2 = πp2/8 cm2

⇒ Area of semi-circle with diameter AC = (π/8) × (q)2 = πq2/8 cm2

⇒ Area of semi-circle with diameter BC = (π/8) × (p2 + q2) = π(p2 + q2) /8 cm2

∴ Area of shaded region = (pq/2) + (πp2/8) + (πq2/8) – = (pq/2) cm2

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