In the given figure, ABCD is a rectangle inscribed in a circle. If two adjacent sides of the rectangle be 8 cm and 6 cm, calculate :

(i) the radius of the circle; and

(ii) the area of the shaded region.

(i) Given,

Rectangle sides = 8 cm and 6 cm.

Let AB = 8 cm and BC = 6 cm

By pythagoras theorem,

AC² = AB² + BC²

AC² = 8² + 6²

AC² = 64 + 36

AC² = 100

AC = $\sqrt{100}$ = 10 cm.

Diameter of circle = 10 cm

Radius = $\dfrac{10}{2}$ = 5 cm.

Hence, radius of circle = 5 cm.

(ii) Shaded area = Area of circle - Area of rectangle

Area of circle = πr²

= 3.14 × 5²

= 3.14 × 25 = 78.5 cm².

Area of rectangle = 8 × 6 = 48 cm².

Shaded area = 78.5 - 48 = 30.5 cm².

Hence, shaded area = 30.5 cm².