In the figure $ P O \perp Q 0 $. The tangents to the circle at $ P $ and $ Q $ intersect at a point $ T $. Prove that $ P Q $ and $ O T $ are right bisectors of each other. "

Given:

In the figure \( P O \perp Q 0 \). The tangents to the circle at \( P \) and \( Q \) intersect at a point \( T \).

To do: We have to prove that \( P Q \) and \( O T \) are right bisectors of each other.

Solution:

$PT$ and $QT$ are tangents to the circle.

$PT = QT$

$PO\ \perp\ QO$

$OP$ and $OQ$ are radii of the circle and $\angle POQ = 90^o$

$OQTP$ is a square where $PQ$ and $OT$ are diagonals.

Diagonals of a square bisect each other at right angles.

$PQ$ and $OT$ bisect each other at right angles.

Therefore, $PQ$ and $QT$ are right bisectors of each other.

Hence proved.

Related Articles In the figure, common tangents \( P Q \) and \( R S \) to two circles intersect at \( A \). Prove that \( P Q=R S \)."\n
In the figure, \( O Q: P Q=3: 4 \) and perimeter of \( \Delta P O Q=60 \mathrm{~cm} \). Determine \( P Q, Q R \) and \( O P \)."\n
In the figure, \( P Q \) is a tangent from an external point \( P \) to a circle with centre \( O \) and \( O P \) cuts the circle at \( T \) and \( Q O R \) is a diameter. If \( \angle P O R=130^{\circ} \) and \( S \) is a point on the circle, find \( \angle 1+\angle 2 \)."\n
Two circles touch externally at a point \( P \). From a point \( T \) on the tangent at \( P \), tangents \( T O \) and TR are drawn to the circles with points of contact \( Q \) and \( R \) respectively. Prove that \( TQ = TR \)."\n
In the figure, \( P Q \) is tangent at a point \( R \) of the circle with centre \( O \). If \( \angle T R Q=30^{\circ} \), find \( m \angle P R S \)."\n
In the figure, tangents \( P Q \) and \( P R \) are drawn from an external point \( P \) to a circle with centre $O$, such that \( \angle R P Q=30^{\circ} . \) A chord \( R S \) is drawn parallel to the tangent \( P Q \). Find \( \angle R Q S \)."\n
\( A \) is a point at a distance \( 13 \mathrm{~cm} \) from the centre \( O \) of a circle of radius \( 5 \mathrm{~cm} \). \( A P \) and \( A Q \) are the tangents to the circle at \( P \) and \( Q \). If a tangent \( B C \) is drawn at a point \( R \) lying on the minor arc \( P Q \) to intersect \( A P \) at \( B \) and \( A Q \) at \( C \), find the perimeter of the \( \triangle A B C \).
In Fig., $PQ$ is a chord of length $8\ cm$ of a circle of radius $5\ cm$ and center $O$. The tangents at $P$ And $Q$ intersect at point $T$. Find the length of $TP$."\n
Diagonals of a trapezium PQRS intersect each other at the point \( O, P Q \| R S \) and \( P Q=3 \mathrm{RS} \). Find the ratio of the areas of triangles POQ and ROS.
In the figure, there are two concentric circles with centre O. \( P R T \) and \( P Q S \) are tangents to the inner circle from a point \( P \) lying on the outer circle. If \( P R=5 \mathrm{~cm} \), find the length of \( P S \)."\n
In the figure, \(P A \) and \( P B \) are tangents from an external point \( P \) to a circle with centre \( O \). \( L N \) touches the circle at \( M \). Prove that \( P L+L M=P N+M N \)."\n
If p, q are real and p≠q, then show that the roots of the equation $(p-q)x^2+5(p+q)x-2(p-q)=0$ are real and unequal.
In a triangle \( P Q R, N \) is a point on \( P R \) such that \( Q N \perp P R \). If \( P N \). \( N R=Q^{2} \), prove that \( \angle \mathrm{PQR}=90^{\circ} \).
Given that \( \frac{4 p+9 q}{p}=\frac{5 q}{p-q} \) and \( p \) and \( q \) are both positive. The value of $\frac{p}{q}$ is
PQ is a chord of length 4.8 cm of a circle of radius 3 cm. The tangents at P and Q intersect at a point T as shown in the figure. Find the length of the TP."\n
Kickstart Your Career
Get certified by completing the course

Get Started