If the sides of a quadrilateral touch a circle, prove that the sum of a pair of opposite sides is equal to the sum of the other pair.

Given:

The sides of a quadrilateral touch a circle.

To do:

We have to prove that the sum of a pair of opposite sides is equal to the sum of the other pair.

Solution:

Let the sides of a quadrilateral $PQRS$ touch the circle at $A, B, C$ and $D$ respectively.

Proof:

$PA$ and $PD$ are the tangents to the circle from $P$.

This implies,

$PA = PD$..….(i)

Similarly,

$QA = QB$……(ii)

$RC = RB$.….(iii)

$SC = SD$.….(iv)

Adding equations (i), (ii), (iii) and (iv), we get,

$PA + QA + SC + RC = RB + QB + SD + PD$

$PQ + RS = RQ + PS$

Hence proved.

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

40 Views

Kickstart Your Career

Get certified by completing the course

Get Started