A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If $A D$ is perpendicular to $D C$ , $BC = 30$ cm and $BS = 24$ cm, then find the length $D C$ .

$Circle inscribed in quadrilateral ABCD$

Question

A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If

A D

is perpendicular to

D C

,

BC = 30

cm and

BS = 24

cm, then find the length

D C

.

$Circle inscribed in quadrilateral ABCD$

[IMAGE_1]

Accepted Answer

Step 1: Join OP and OQ.
$BR = BS = 24$ cm
$	herefore CR = 6$ cm
Step 2: $\Rightarrow CQ = 6$ cm
Step 3: Also, $DQ = OP = 8$ cm (radius, since $AD \perp DC$)
Step 4: Hence, $DC = DQ + QC = 8 + 6 = 14$ cm

Question

Solution