The quadrilateral $ABCD$ is circumscribed in the figure. At $P,Q,R,$, the circle touches the sides $AB,BC,CD,DA.$ Find the perimeter of a quad  $ABCD$ if $AP=9cm,BP=7cm,CQ=5cm,DR=6cm.$

Assume a circle with centre 'O' that is circumscribed by quadrilateral ABCD that touches the circle at four points, and then we will use the property of tangents of a circle that the lengths of tangents drawn from the external points are equal.  Perimeter of the quadrilateral is adding all sides

ABCD is a quadrilateral and let O be the center of the circle inscribed in the quadrilateral.

$\mathrm{AB},\mathrm{BC},\mathrm{CD}$ and $\mathrm{AD}$are the tangent to the circle at $\mathrm{P},\mathrm{Q},\mathrm{R}$ and $\mathrm{S},\mathrm{AP}=9$ ,$PB=7, QC=5$ and $\mathrm{DR}=6$

We know that the lengths of the two tangents from a point to circle are equal.

Thus, $\mathrm{AP}=\mathrm{AS}=9 \mathrm{cm}$

$\text{BP}=\text{BQ}=7 \text{cm}$

$\text{CQ}=\text{CR}=5 \text{cm}$

$\text{DR}=\text{DS}=6 \text{cm}$

Perimeter of quadrilateral $\mathrm{ABCD}=\mathrm{AB}+\mathrm{BC}+\mathrm{CD}+\mathrm{AD}$

$AS=9+7+7+5+5+6+6+9=54 \text{cm}$