Question

$ABCD$ is a quadrilateral such that all of its sides touch a circle. If $AB=6cm,BC=6.5cm$ and $CD=7cm$, then find the length of $AD$

Answer 1

Let $P,Q,R$ and $S$ be the points where the circle touches the quadrilateral. We know that the lengths of the two tangents drawn from an exterior point to a circle are equal.
Thus, we have, $AP=AS,BP=BQ,CR=CQ$ and $DR=DS$.
Hence, $AP+BP+CR+DR=AS+BQ+CQ+DS$
$\Rightarrow AB+CD=AD+BC$.
$\Rightarrow AD=AB+CD-BC$
$=6+7-6.5=6.5$
Thus, $AD=6.5cm$.