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AB and CD are two perpendicular diameters of a circle with center O, OD is the

diameter of smaller circle. If $OA = 7 cm,$ then the area of the shaded region in figure, is,

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66.5 {cm}^{2}

66 {cm}^{2}

65 {cm}^{2}

67.5 {cm}^{2}

answer is A.

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Detailed Solution

Given,
Radius of bigger circle

r_{1} (OA) = 7

Thus, its diameter
AB = 2R = 14 cm
Radius of smaller circle

r_{2},

r_{2} = \frac{r_{1}}{2}

r_{2} = \frac{7}{2}

Now calculate area of the bigger circle

C_{1}

C_{1} = π r_{1}^{2}

C_{1} = \frac{22}{7} \times 7 \times 7

C_{1} = 154 {cm}^{2}

So, area of the semicircle is,

= \frac{154}{2} {cm}^{2}

= 77 {cm}^{2}

Area of the shaded smaller circular region

C_{2}

C_{2} = π r_{2}^{2}

C_{2} = \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2}

C_{2} = \frac{77}{2}

C_{2} = 38.5

Now, total area of unshaded triangle,

△ ABC = \frac{1}{2} \times AB \times OC

= \frac{1}{2} \times 14 \times 7

= 49 {cm}^{2}

Area of the shaded region =Area of the shaded smaller circular region + (Area of semicircle -
Area of the

△ ABC)

= 38.5 + (77 - 49)

= 66.5 {cm}^{2}

Correct option is 1.

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