Find the perimeter of the triangle.

Question

Point $P$ lies outside circle $O$. Two tangents $PM$ and $PN$ are drawn ($M$ and $N$ lie on $O$) and a third tangent to O intersects segments $PM$ and $PN$ at $Q$ and $R$ respectively. Note that since it says segments and not lines, it means that the point $Q$ lies between $P$ and $M$ and the point $R$ lies between $P$ and $M$. If $PM=10$, find the perimeter of triangle $PQR$

Answer 1

Let $QM$ be $x$ and $RN$ be $y$. Then $PQ = 10-x$ and $PR = 10-y$. ($PR = 10$ because of tangential properties).
Also, let $S$ be the point where the third tangent touches the circle. Again, $QM=QS=x$ and $RN=RS=y$ because of tangential properties. Finally, sum up the sides $PQ$, $PR$ and $QR$ and you get 20.

Answer 2

$$PM=PN=10 \ cm$$ Let $PR=x$ So $RN=10-x$ Simlarly $$PQ=y$$ and $$QM=10-y$$ It can be proved that $$QB=QM$$ and$$BR=RN$$ So$$Perimeter=10-x+x+10-y+y=20 \ cm $$