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In fig., AB and CD are two diameters of a circle O, which are perpendicular to each other. OB is the diameter of the smaller circle. If $OA=7\ cm$, find the area of the shaded region.[use $\pi =\frac{22}{7}$].
Given: A larger circle with center O and diameter AB and CD. OB is the diameter of the smaller circle. and $OA=7\ cm$.
To do: To find the area of the shaded region
Solution:
AB and CD are the diameters of a circle with centre O. $( Radius\ of\ the\ circle)$
$OA=OB=OC=OD=7\ cm$
Area of the circle with diameter $AB=2\pi r^{2}$
$=2\times \frac{22}{7} \times 7\times 7$
$=308\ cm^{2}$
Another shaded circle with diameter $OB=7\ cm$
Radius of shaded circle$=\frac{7}{2}\ cm$
Area of the shaded circle with diameter $OB=\pi r^{2}$
$=2\times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2}$
$=\frac{77}{2} \ cm^{2}$
Area of $\vartriangle ACD=\frac{1}{2} \times base\times height$
$=\frac{1}{2} \times 14\times 7$
$=49\ cm^{2}$
Area of semi-circle with diameter AB,
$=\frac{308}{2}$
$=154\ cm^{2}$
Therefore area of the shaded region$=$Area of the circle with diameter OB$+$Area of the semi-circle-area of $\vartriangle ACD$
$=\frac{77}{2} +154-49$
$=66.5\ cm^{2}$
Therefore, Area of the shaded region is $66.5\ cm^{2}$ .
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