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In the given figure, APB and AQO are semicircles and AO = OB. If the perimeter of the figure is 40 cm, find the area of the shaded region.
AO = OB
Perimeter of the figure = 40 cm………….. (i)
Let the diameters of semicircles AQO and APB be ‘x1’ and ‘x2’ respectively.
Then, using (1), we have
AO = OB
Also, AB = AO + OB = AO + AO = 2AO
⇒ x2 = 2x1
So, diameter of APB = 2x1
and diameter of AQO = x1
Radius of APB = x1
and Radius of AQO = ………….. (ii)
Perimeter of shaded region = perimeter of AQO + perimeter APB + diameter of APB ………………… (iii)
∵ Perimeter of semicircle = πr
∴ Perimeter of semicircle AQO = × = cm
Perimeter of semicircle APB = × x1 = cm
Now, using (iii), we have
40 = + + x1
40 × 7 = 40x1
280 = 40x1
x1 = = 7 cm
∴ using (ii), we have
Radius of APB = 7 cm = r1
And Radius of AQO = cm = 3.5 cm = r2
∵ Area of semicircle = πr2
∴ Area of semicircle APB = πr12
= × × 7 × 7 = 11 × 7 = 77 cm2
Area of semicircle APB = πr22
= × × 3.5 × 3.5 = 19.25 cm2
Thus, Area of shaded region = Area of APB + Area of AQO
= 77 + 19.25 = 96.25 cm2
Hence, the area of the shaded region is 96.25 cm2.