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In Fig. 12.27, AB and CD are two diamete...

In Fig. 12.27, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If`O A=7\ c m`, find the area of the shaded region

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We use the concepts of areas of circles and areas of triangles to solve the problem.
AB and CD are diameters of the circle with center O
∴ OD = OC = OA = OB =
Radius of the circle R = 7 cm
∴ AB = 2R = 14 cm
Radius of shaded circular region, r = OD/2 = 7/2 cm
Area of the shaded smaller circular region = `pi r^@`
= `pi (7/2 cm)^2`
= `22/7 × 7/2 × 7/2 cm^2`= `77/2 cm^2`= `38.5 cm^2`
Area of the shaded segment of larger circular region = Area of semicircle ACB - Area of ΔABC
= `1/2 pi(OA)^2 - 1/2 × AB × OC`= `1/2 pi R^2 - 1/2 × 2R × R`
= `1/2 × 22/7 × 72 - 1/2 × 14 × 7`
= `77 - 49`
= `28 cm^2`
Area of the shaded region = Area of the shaded smaller circular region + Area of the shaded segment of larger circular region
= `38.5 cm^2 + 28 cm^2`
= `66.5 cm^2`
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