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Question

In the figure below, $ \mathrm{AB}$ and $ \mathrm{CD}$ are two diameters of a circle (with center $ \mathrm{O}$) perpendicular to each other, and $ \mathrm{OD}$ is the diameter of the smaller circle. If $ \mathrm{OA} = 7 \mathrm{cm}$, find the area of the shaded region.

A

66.5cm2

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B

66cm2

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C

65cm2

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D

67.5cm2

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Solution

The correct option is A

66.5cm2


The explanation for the correct answer.

Step 1: Find the radius and area of the circle with the diameter OD

Given: Radius of the circle OA=7cm=OD.

Radius of circle =OD2=72cm

Area of the circle =πr2

=227×72×72=38.5cm2

Step 2: Find the area of the semi-circle with a diameter of AB

The radius of the semi-circle =7cm

Area of the semi-circle =12×πr2

=12×227×7×7=77cm2

Step 3: Find the area of the triangle BCA

The base of the triangle AB=7+7=14cm

Height of the triangle OC=OD=7cm

Area of the triangle =12×b×h

=12×14×7=49cm2

Step 4: Area of the shaded region

Area of the shaded region =Area of the circle with diameter OD +Area of the semi-circle with diameter AB -Area of the triangle ABC

Area of the shaded region =38.5+77-49

=66.5cm2

Hence option A is the correct answer.


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