Question

OABC is a rhombus whose three vertices A, B and C lie on a circle with centre O.

(i) If the radius of the circle is 10 cm, find the area of the rhombus.

(ii) If the area of the rhombus is $32\sqrt{3}c{m}^2$ find the radius of the circle.

Answer 1

Given OABC is a rhombus whose three vertices A, B, C lies on circle with centre O and radius 10 cm.

Suppose the diagonals of the Rhombus OABC intersects at S.

Radius of circle(r) = 10 cm.

∴ OA = OB = OC = 10 cm.

Diagonals of rhombus bisect each other at 90^o.

∴ OS = SB = $\frac{OB}{2}=\frac{10}{2}$ = 5 cm and SC = SA

In a right angle triangle, OCS,

OC² = OS² + SC²

⇒ (10)² = 5² + SC²

⇒ SC² = 100 – 25

⇒ SC² = 75

⇒ $SC=5\sqrt{3}cm$

∴$AC=2\times SC=2\times 5\sqrt{3}=10\sqrt{3}cm$

Area of Rhombus = $\frac{1}{2}\times d_1\times d_2$

= $\frac{1}{2}\times 10\times 10\sqrt{3}$
= $50\sqrt{3}$ cm²

∴ Area of Rhombus = $50\sqrt{3}$ cm²

^{(ii)Area of the rhombus =} $32\sqrt{3}c{m}^2=\frac{1}{2}\times d_1\times d_2{d}_1=radiusand{d}_2=\sqrt{r^2-(\frac{r}{2}{)}^2}{d}_2=\frac{\sqrt{3}}{2}r$

$32\sqrt{3}c{m}^2=r\times \frac{\sqrt{3}}{2}r{r}^2=64r=8cm$