# 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 ‘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 =

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

Now,

Area of semicircle = πr2

Area of semicircle APB = πr12

= × × 7 × 7 = 11 × 7 = 77 cm2

Similarly,

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.

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