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Find the area of the shaded region in Fi...

Find the area of the shaded region in Fig. 12.19, if `P Q=24\ c m ,\ P R=7\ c m` and O is the centre of the circle.

Text Solution

Verified by Experts

We know that, the angle in a semicircle is a right angle.
∴ `/_RPQ = 90^@`
Thus, `triangleRQP` is a right-angled triangle.
Given, PQ = `24` cm, PR = `7` cm
∴ Using Pythagoras theorem,
`RQ^2 = PR^2 + PQ^2`
RQ = `sqrt(7^2 + 24^2)`
= `sqrt(49 + 576)`
= `sqrt(625)`
Thus, PQ = `25` cm which is the diameter
∴ Radius (r) = `25/2` cm
Area of shaded region = Area of semicircle RPQ - Area of triangleRQP
= `1/2 × pir^2 - 1/2 xx PQ xx RP`= `1/2 [(22/7 xx 25/2 xx 25/2) - (24 xx 7)]`
= `1/2 [6875/14 - 168]`
= `1/2 [(6875 - 2352)/14]`
= `1/2 xx 4523/14`
= `4523/28 cm^2`
= `161.54 cm^2 `(approximately)
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