Question 12 Marks
Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral.
Answer
View full question & answer→Given two tangents PQ and PR are drawn from an external point to a circle with centre O.
To prove QORP is a cyclic quadrilateral.
Proof: since, PR and PQ are tangents.
So, OR $\perp$ PR and OQ $\perp$ PQ
[since, if we draw a line from centre of a circle to its tangent line. Then, the line always perpendicular to the tangent line]
$\therefore\ \angle\text{ORP}=\angle\text{OQP}=90^\circ$
Hence, $\angle\text{ORP}+\angle\text{OQP}=180^\circ$
So, QORP is cyclic quadrilateral.
[If sum of opposite angles is quadrilateral in 180°, then the quadrilateral is cyclic]
Hence proved.
To prove QORP is a cyclic quadrilateral.
Proof: since, PR and PQ are tangents.
So, OR $\perp$ PR and OQ $\perp$ PQ
[since, if we draw a line from centre of a circle to its tangent line. Then, the line always perpendicular to the tangent line]
$\therefore\ \angle\text{ORP}=\angle\text{OQP}=90^\circ$
Hence, $\angle\text{ORP}+\angle\text{OQP}=180^\circ$
So, QORP is cyclic quadrilateral.
[If sum of opposite angles is quadrilateral in 180°, then the quadrilateral is cyclic]
Hence proved.