Two tangents $PQ$ and $PR$ are drawn from an external point $P$ to a circle with centre $O$. Prove that $QORP$ is a cyclic quadrilateral.

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(N/A) Given: Two tangents $PQ$ and $PR$ are drawn from an external point $P$ to a circle with centre $O$.
To prove: $QORP$ is a cyclic quadrilateral.
Proof:
$1$. Since $PQ$ and $PR$ are tangents to the circle at points $Q$ and $R$ respectively,the radius is perpendicular to the tangent at the point of contact.
$2$. Therefore,$OQ \perp PQ$ and $OR \perp PR$.
$3$. This implies $\angle OQP = 90^{\circ}$ and $\angle ORP = 90^{\circ}$.
$4$. In quadrilateral $QORP$,the sum of the opposite angles is $\angle OQP + \angle ORP = 90^{\circ} + 90^{\circ} = 180^{\circ}$.
$5$. Since the sum of a pair of opposite angles in the quadrilateral $QORP$ is $180^{\circ}$,it is a cyclic quadrilateral.
Hence proved.

Explore More

Similar Questions

$\square PQRS$ is a cyclic quadrilateral. If $m \angle P = 30^{\circ}$, then $m \angle R = \dots$ (in $^{\circ}$)

Difficult
View Solution

If an isosceles triangle $ABC$, in which $AB = AC = 6\, cm$, is inscribed in a circle of radius $9\, cm$, find the area of the triangle in $cm^{2}$.

Difficult
View Solution

The maximum area of a triangle inscribed in a semicircle with diameter $50 \, cm$ is ........... $cm^{2}$.

Write 'True' or 'False' and give reasons for your answer.
In the figure,$PQL$ and $PRM$ are tangents to the circle with center $O$ at the points $Q$ and $R$ respectively,and $S$ is a point on the circle such that $\angle SQL = 50^{\circ}$ and $\angle SRM = 60^{\circ}$. Then $\angle QSR$ is equal to $40^{\circ}$.

In the figure,tangents $PQ$ and $PR$ are drawn to a circle such that $\angle RPQ = 30^{\circ}$. $A$ chord $RS$ is drawn parallel to the tangent $PQ$. Find the $\angle RQS$. (in $^{\circ}$)

Difficult
View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo