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 \[10\text{ }cm\], BC \[=38\text{ }cm\], PB \[=27\text{ }cm\]and AD\[\bot
\]CD, then find the length of CD.

Hint:As given in the diagram, first we need to find the length of QC as BQ is equal to BP due to congruency and after finding the length we add the length with the value of RD to find the value of CD. Both SD and RD are equal due to AD\[\bot \]CD.

Complete step by step solution:
First we write the values given in the question such as BC \[=38\text{ }cm\], PB \[=27\text{ }cm\]and due to the radius of the circle being \[10\text{ }cm\]. The value of OP and OS are \[10\text{ }cm\] also.

Now the values of the line BP and BQ are the same due to the circle touching the point of the quadrilateral at the same distance that is the radius of the circle i.e. \[10\text{ }cm\]. Hence, with the value of BP or PB as
\[27\text{ }cm\], the length of BC is given as \[27\text{ }cm\] also.

Now using the same congruence method that tells that all the points of the circle that are forming a tangent with the quadrilateral are equal making the value of CQ equal to CR. With BC \[=38\text{ }cm\]
and BQ equal to PB we get the value of QC as:
QC \[=\] BC \[-\] BQ
QC \[=\] \[38-27=11\]cm

Now as proved above both QC and CR are equal and to find the value of CD we need to add the value of CR and RD as:
CD \[=\] CQ \[+\] RD

The value of RD is equal to the radius of the circle i.e. \[10\text{ }cm\] as it forms a rectangle with two sides AD and CD being at \[{{90}^{\circ }}\].

CD \[=\] \[11+10=21\]cm

Therefore, the value of CD is given as \[21\] cm.

Note:The difference between a square and a quadrilateral is that those in quadrilateral sides and angles are same but not all of them are equal otherwise it will be a square. Here although the diagram looks like a square but it is not a square and doesn’t follow its properties as one corner of it is acting like a rectangle while other like a square.