If line PQ is tangent to circle R at point Q, and line PS is tangent to ⊙R at point S, what is the perimeter of quadrilateral PQRS?

Question

Assuming that 4 refers to line $T$, would that not make this question impossible since $T$ is the hypotenuse of $\overline{PQ}$ and $\overline{RQ}$?

If $4$ were to be the length of $\overline{PQ}$, then none of the answers would match.

Answer 1

$PR = 4 + 5 = 9$ so $PQ = \sqrt{9^2 - 5^2} = \sqrt{56} = 2 \sqrt{14}$ as $PQ \perp QR$. Now as tangents from a common point have the same length, the perimeter is $2 \sqrt{14} \cdot 2 + 5 \cdot 2 = 10 + 4 \sqrt{14}$, so your answer is correct.