Find the perimeter and area of the shaded region in the given figure.

Given,

AB = 12 cm

BC = 16 cm

AC is the diameter of the semicircle.

Angle in a semicircle is a right angle, thus, ABC = 90°

Using Pythagoras theorem in △ABC:

⇒ AC² = AB² + BC²

⇒ AC² = 12² + 16²

⇒ AC² = 144 + 256

⇒ AC² = 400

⇒ AC = $\sqrt{400}$ = 20 cm.

So, radius = $\dfrac{20}{2}$ = 10 cm.

Semicircle arc = πr = 3.14 × 10 = 31.4 cm

Perimeter = Semicircle arc + side AB + side BC

Perimeter = 31.4 + 12 + 16

= 59.4 cm

Area of shaded region = Area of semicircle - Area of triangle

Calculating the area of semicircle,

$\text{Area of semicircle} = \dfrac{1}{2} πr^2 \\[1em] = \dfrac{1}{2} \times 3.14 \times 10^2 \\[1em] = \dfrac{1}{2} \times 3.14 \times 100 \\[1em] = 3.14 \times 50 \\[1em] = 157 \text{ cm}^2.$

Area of triangle = $\dfrac{1}{2}$ × AB × BC

= $\dfrac{1}{2}$ × 12 × 16

= 6 × 16 = 96 cm².

Area of shaded region = 157 - 96 = 61 cm².

Hence, perimeter of shaded region = 59.4 cm and area of shaded region = 61 cm².