Question

In the given figure ABCD is a quadrilateral in which $D={90}^{\circ }$. A circle C(O,r) touches the sides AB , BC , CD and DA at P , Q , R and S, respectively. If BC = 38cm , CD = 25 cm and BP =27cm then find the value of r.

Answer 1

Given: ABCD is a quadrilateral such that $\angle D={90}^{\circ }$
BC = 38 cm, CD = 25 cm and BP = 27 cm
$\therefore$ From the figure,
BP = BQ = 27 cm [Tangents from an external point]
BC = 38
$\Rightarrow BQ+QC=38$
$\Rightarrow 27+QC=38$
$\Rightarrow QC=38-27$
$\Rightarrow QC=11cm$
$\therefore QC=11cm=CR$ [Tangents from an external point]
CD = 25 cm
CR + RD = 25
$\Rightarrow 11+RD=25$
$\Rightarrow RD=25-11$
$\Rightarrow RD=14cm$
Also,
RD = DS = 14 cm
$\therefore$ OR and OS are radii of the circle
From tangents R and S, $\angle ORD=\angle OSD={90}^{\circ }$
Now, ORDS is a square
$\therefore$ OR = DS = 14 cm
Thus, radius, r = 14 cm