Question

In the figure, a quadrilateral $PQRS$ is drawn to circumscribe a circle. Prove that $PQ+RS=PS+QR$

Answer 1

Given:

Quadrilateral $PQRS$ circumscribes a circle

To prove that:

$PQ+RS=PS+QR$

Proof:

Lengths of tangents drawn to a circle from an external point are equal.

Hence, $PA=PD...........\left(i\right)$

$QA=QB...........\left(ii\right)$

$RB=RC...........\left(iii\right)$

$SC=SD...........\left(iv\right)$

LHS

$PQ+RS=PA+AQ+RC+CS...............\left(v\right)$

RHS

$PS+QR=PD+DS+QB+BR...............\left(vi\right)$

$=PA+CS+AQ+RC$ using equations (i), (ii), (iii) and (iv)

From above, it can be seen LHS = RHS

Hence proved