Question

The parallelogram circumscribing a circle is a rhombus.

• A. True
• B. False
• C. Either
• D. Neither

Answer 1

The correct option is A True

Consider the above drawn parallelogram.

We know that the tangents drawn to a circle from an exterior point are equal in length.

Therefore, $AP=AS,BP=BQ,CR=CQ$ and $DR=DS$

Adding the above equations,

$AP+BP+CR+DR=AS+BQ+CQ+DS\Rightarrow (AP+BP)+(CR+DR)=(AS+DS)+(BQ+CQ)\Rightarrow AB+CD=AD+BC\Rightarrow 2AB=2BC$

(Since $ABCD$ is a parallelogram, thus, $AB=DC$ and $AD=BC$)

$AB=BC$

Therefore, $AB=BC=DC=AD$.

Hence, $ABCD$ is a rhombus.

Hence, the given statement the parallelogram circumscribing a circle is a rhombus is true.