$S$ and $Q$ are situated in opposite regions with respect to $PR$ in $\triangle PQR$ ,What is the length of $QS$?

Question

I am currently working on an Olympiad math problem, and I am struggling to find a solution. I would greatly appreciate your help in solving this problem. I was unable to solve the problem because I couldn't use this condition of the length of $PS = 20$ and $ST = 15$. Besides, I don't understand What possible strategies or formulas can be used to solve this problem? Then how can I suppose to get the value of $QS$.

A small hint will be enough for me to proceed.

Source: Bangladesh Math Olympiad (BDMO)

In triangle $\triangle PQR$, $\angle R= 90^\circ$ and $QR = 21$. $T$ is a point on the side $PR$ such that $RT = 47$. $S$ and $Q$ are situated in opposite regions with respect to $PR$ in such a way so that $\angle PST = 90^\circ$ .If $PS = 20$, $ST = 15$ find the length of $QS$.

Answer 1

1. PT = … = 25. Then, RP = 47 + 25.

2. QP = … = 75.

3. Find $\alpha$ and $\beta$.

4. Apply cosine law to ......

Answer 2

Hint: $PT=25$ (Pythagorean triple). Extend $QR$ and project $S$ onto $QR$. Let's say $V$ is the projected point. To find $QS$, we need to find $VS$ and $VR$. Project $T$ onto line $RS$ and use similarity of triangles $PST$ and the newly formed right triangle.