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In the figure given below O is the centre of the circle and OPQR is a rectangle, A is a point on PO such that $AO = \frac{1}{3} PO$ and B is the mid-point of OR. Find the area of the shaded region, if PA = 8 cm and BR = 4cm $(Use π = 3.14) .$

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132.68 {cm}^{2}

121.12 {cm}^{2}

108.56 {cm}^{2}

None of these

answer is B.

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Detailed Solution

Given,

AO = \frac{1}{3} PO

Also, B is the mid-point of OR,

PA = 8 cm

BR = 4 cm

From figure,

PO = PA + AO

PO = PA + \frac{PO}{3}

\frac{2 PO}{3} = PA = 8

PO = 12 cm

Now again from figure,

OR = OB + BR = 2 BR

OR = 2 \times 4

OR = 8 cm

OB = BR = 4 cm

Now, the area of shaded region,
A= area of rectangle OPQR +

\frac{3}{4}

(area of circle)-

\frac{1}{4}

(area of circle)

A = PO \times OR + \frac{3}{4} π {(OB)}^{2} - \frac{1}{4} π {(OB)}^{2}

A = 12 \times 8 + \frac{3}{4} π {(4)}^{2} - \frac{1}{4} π {(4)}^{2}

A = 96 + 3 \times 3.14 \times 4 - 3.14 \times 4

A = 121.12 {cm}^{2}

This is the required area.
Correct option is 2

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