Question

A circle is inscribed in a quadrilateral $ABCD$ touching its sides $AB,BC,CD,AD$ at $P,Q,R$ and $S$ respectively. If the radius of the circle is $10cm,BC=38cm,PB=27cm$ and $AD\perp CD$, then find the length of $CD$.

Answer 1

$OP=OS=10cm$

$BC=38cm$

$PB=27cm$

$AD\perp CD$

Since, tangent segment to circle from extant point are congruent

$Bp=BQ=27cm$

$CQ=CR$

$BC=28cm$

$BQ+QC=38$

$QC=38-27=11cm$

Since all angle in Quad $DROS$ are right angle

$DROS$ is a rectangle

$OS=RD=10cm$

$CD=CR+RD$

$=CQ+RD$

$=11+10\Rightarrow CD=21cm$