##### In the given figure, ΔABC is right-angled at A. Semicircles are drawn on AB, AC and BC as diameters. It is given that AB = 3 cm and AC = 4 cm. Find the area of the shaded region.

Here we will first find out the area of semicircle whose diameter is BC and then subtract the area of right angle triangle ABC from it and then we will subtract this result from the area of semicircles whose diameters are AB and AC.

Consider ∆ABC, BAC = 90°

BC2 = 42 + 32

BC2 = 16 + 9

BC2 = 25

BC = 5 cm

Area of semicircle whose diameter is AC

Area of semicircle = 2π cm2 –eqn2

Area of semicircle whose diameter is AB

Area of semicircle = 1.125π cm2 eqn3

Area of semicircle whose diameter is BC

Area of semicircle = 3.125π cm2 eqn4

Area of triangle PQR = 3×2

Area of triangle PQR = 6 cm2 eqn5

Now subtract equation 5 from equation 4,

Area of semicircle excluding ∆ABC = eqn4 – eqn5

Area of semicircle excluding ∆ABC = 3.125π – 6

Area of semicircle excluding ∆ABC = 3.8214 cm2 eqn6

Area of shaded region = eqn3 + eqn2 – eqn6

Area of shaded region = 2π + 1.125π – 3.8214

Area of shaded region = 3.125π – 3.8214

Area of shaded region = 9.8214 – 3.8214

Area of shaded region = 6 cm2

Area of shaded region 6 cm2.

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