In the figure, l(AB) = 14 cm. If the diameters of the semicircles AM and MB are equal, what is the total area of the shaded part?

Given: AB = 14 cm & AM = MB

Clearly, AB = AM + MB

⇒ AB = AM + AM = 2 AM

⇒ AM = AB/2

⇒ AM = 14/2 = 7 cm

Also, area of circle is given by

Area = πr²

Area of semicircle = πr²/2

Now, if diameter of one of the semicircle = 7 cm

Then, radius of that semicircle = 7/2 = 3.5 cm

So, area of that semicircle

…(i)

Radius of the other semicircle = 7/2 = 3.5 cm

So, area of that other semicircle

…(ii)

Adding equations (i) and (ii), we get

Total area of the shaded part = 19.25 + 19.25

= 38.5 cm

Thus, total area of the shaded part is 38.5 cm².

Alternate Method:

Given that AB = 14 cm & AM = MB

Notice, AB = AM + MB

⇒ AB = AM + AM

⇒ AB = 2 AM

⇒ AM = AB/2

⇒ AM = 14/2 = 7 cm

So, if AM is joined with MB, then it forms a complete circle of diameter, 7 cm.

Then, radius of this circle = 7/2 = 3.5 cm

When points A and B are met together, they form a circle.

Let AB = L and center of this circle = O.

Then, we have LM = 7 cm and LO = OM = 3.5 cm (radius).

Area of this circle is given by

Area = πr²

⇒ Area = 22/7 × 3.5²

⇒ Area = (22 × 3.5 × 3.5)/7

⇒ Area = 269.5/7 = 38.5 cm²

Thus, total area of the shaded part is 38.5 cm².