Question

ABCD is a quadrilateral such that $\angle D={90}^{\circ }$. A circle $C(O,r)$ touches the sides $AB,BC,CD$ and $DA$ at $P,Q,RandS$ respectively. If $BC=38cm,CD=25cm$ and $BP=27cm$, find $r$ (in cm).

Answer 1

Given: $BP=27cm,BC=38cm$ and $CD=25cm$
$BP=BQ$ [$\because$ Length of tangents from external point are equal]
So, $BQ=27cm$
But $BC=BQ+QC$
$38=27+QC$
$QC=11cm$
Now $QC=CR$ [$\because$ Length of tangent from external point are equal]
$CR=11cm$
But $CD=CR+RD$
$25=11+RD$
$RD=14cm$
Now, ORDS is a square
So, $OR=OD$
$OR=14cm$
$\therefore$ Radius of circle $=14cm$