Name: Tangents From an Exterior Point: Illustration
Uploaded: 2019-07-19
Duration: 4 min 47 s
Description: If you lift a circular object with a pair of tongs, then the tongs are two tangents to the circular object. Let us study the properties of tangents drawn fro...

Question

Prove that the parallelogram circumscribing a circle is a rhombus.

Accepted Answer

Let $ABCD$ be a quadrilateral circumscribing a circle with centre $O$ and touching at points $P, Q, R, S$ .

We know that, tangents drawn from external point to circle are equal.
SO,
$AP = AS - - - - - (i)$
$BP = BQ - - - - - (ii)$
$CQ = CR - - - - - (iii)$
$DR = DS - - - - - (iv)$

Adding all the equations on both sides,
$\Rightarrow AP + BP + CR + DR = AS + BQ + CQ + DS$

$\Rightarrow AB + CD = BC + AD - - - - - (i)$

In a parallelogram $ABCD$ .
$AB = CD; AD = BC$

From $(i)$ ,
$\Rightarrow AB + AB = BC + BC$
$\Rightarrow AB = BC$

So, $AB = BC = CD = AD$

We know that, a parallelogram with all equal sides are known as rhombus.

Therefore, parallelogram circumscribing a circle is rhombus.

Hence, proved.

Prove that the parallelogram circumscribing a circle is a rhombus.

Important Questions on Circles

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