Question

In the given figure, OPQR is a rhombus, three of whose vertices lie on a circle with centre O. If the area of the rhombus is `32sqrt(3)`, find the radius of the circle.

Answer 1

In a rhombus, all sides are congruent to each other.

Thus, we have:

`"OP" = "PQ" = "QR" = "RO"`

Now, consider ΔQOP. 

OQ  = OP (Both are radii.)

Therefore, ΔQOP is equilateral.

similarly, ΔQOR is also euilateral and ΔQOP ≅  ΔQOR

Ar. (QROP) = Ar.( ΔQOP) + A (ΔQOP) + A(ΔQOR) = 2 Ar. [ΔQOP] 

Ar. (ΔQOP) `=1/2xx32sqrt(3) = 16 sqrt(3)` 

Or,

`16sqrt(3)=sqrt(3)/4"s"^2`  (where s is the side of the rhombus)

Or,

s² = 16 × 4 = 64 

⇒ s = 8 cm

∴ OQ = 8 cm

Hence, the  radius of the circle is 8 cm.