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.

Given:

AO = OB

Perimeter of the figure = 40 cm………….. (i)

Let the diameters of semicircles AQO and APB be ‘x_{1}’ and ‘x_{2}’ respectively.

Then, using (1), we have

AO = OB

Also, AB = AO + OB = AO + AO = 2AO

⇒ x_{2} = 2x_{1}

So, diameter of APB = 2x_{1}

and diameter of AQO = x_{1}

Radius of APB = x_{1}

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 = × x_{1} = cm

Now, using (iii), we have

40 = + + x_{1}

40 =

40 × 7 = 40x_{1}

280 = 40x_{1}

x_{1} = = 7 cm

∴ using (ii), we have

Radius of APB = 7 cm = r_{1}

And Radius of AQO = cm = 3.5 cm = r_{2}

Now,

∵ Area of semicircle = πr^{2}

∴ Area of semicircle APB = πr_{1}^{2}

= × × 7 × 7 = 11 × 7 = 77 cm^{2}

Similarly,

Area of semicircle APB = πr_{2}^{2}

= × × 3.5 × 3.5 = 19.25 cm^{2}

Thus, Area of shaded region = Area of APB + Area of AQO

= 77 + 19.25 = 96.25 cm^{2}

__Hence, the area of the shaded region is 96.25 cm ^{2}.__

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