<s>[INST] What is the value of $x+y$ if x and y are co-primes and $PR=\dfrac{x}{y}$ in the diagram? 

$ST$ is the perpendicular bisector of $PR$ and $SP$ is the angle
  bisector of $\angle QPR$. If $QS=9cm$ and $SR=7cm$ then
  $PR=\dfrac{x}{y}$ where x, y are co-primes. $x+y$=?

Source: Bangladesh Math Olympiad 2015 junior category. 
I found that $PS=7$, but I can't get the value of $PR$. 
 [/INST] 
As shown, by the angle bisector theorem,
$${(\sqrt{7^2-a^2} + \sqrt{9^2-a^2})\over 9}={(\sqrt{7^2-a^2} + \sqrt{7^2-a^2})\over 7}$$
Simplify,
$$7\sqrt{9^2-a^2}=11\sqrt{7^2-a^2}\implies 7^29^2-49a^2=7^211^2-121a^2\implies 72a^2=$$
$$7^2(11^2-9^2)=7^2\times40\implies a^2={245\over 9}$$
Therefore $PR=2\sqrt{7^2-a^2}=2\sqrt{49-{245\over 9}}=2\sqrt{{196\over 9}}={28\over 3}$
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