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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.
Solution
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.
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