In Fig., ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.

detailed solution

1

here, we have to find the area of semicircle ,

In triangle ABC 

By using Pythagoras

Let the radius are, 

AB = AC = 14 cm 

Using  Pythagoras theorem, we can find the hypotenuse (BC) of ΔABC.

BC² = AB² $+$ AC²

= (14)² $+$ (14)²

BC = $\sqrt{2}$ $\times$ (14)²

= 14$\sqrt{2}$ cm

Therefore, The radius of semicircle BDC, r = BC/2 = 14$\sqrt{2}$/2 = 7$\sqrt{2}$ cm

Area of the shaded region = Area of semicircle $-$ (Area of quadrant ABC $-$ Area ΔABC)

= $\mathrm{\pi }$r²/2 $-$ [90⁰/360⁰ $\times$ $\mathrm{\pi }$(14)² $-$ 1/2 $\times$ AC $\times$ AB]

= $\mathrm{\pi }$(7$\sqrt{2}$)²/2 $-$ [$\mathrm{\pi }$(14)²/4 $-$ 1/2 $\times$ 14 $\times$ 14]

= [(22 $\times$ 7 $\times$ 7 $\times$ 2)/(7 × 2)] $-$ [(22 $\times$ 14 $\times$ 14)/(7 $\times$ 4) - 7 $\times$14]

= 154 $-$ (154 $-$ 98)

= 98 cm²