Correct Answer: $\frac{4}{5}$

Solution :
$\angle$ P = 90°
Pythagoras theorem
Hypotenuse$^2$ = perpendicular$^2$ + base$^2$
Take the sides as shown in the figure
S and T are the midpoints of PR and PQ.
QR$^2$  = QP$^2$  + PR$^2$
⇒ QR$^2$  = 2$^2$  + 2$^2$
⇒ QR$^2$  = 4 + 4 = 8
⇒ QR = $2\sqrt{2}$
QS$^2$  = QP$^2$  + PS$^2$
⇒ QS$^2$  = 2$^2$  + 1$^2$
⇒ QS$^2$  = 4 + 1
⇒ QS = $\sqrt{5}$
RT$^2$  = PT$^2$  + PR$^2$
⇒ RT$^2$  = 1$^2$  + 2$^2$
⇒ RT$^2$  = 5
⇒ RT = $\sqrt{5}$
So, $\frac{\text{RQ}^2}{(\text{QS}^2 + \text{RT}^2)}$
= $\frac{(2\sqrt{2})^2}{[(\sqrt{5})^2 + (\sqrt{5})^2]}$
= $\frac{8}{10}$
= $\frac{4}{5}$
$\therefore$ The value of $\frac{\text{RQ}^2}{(\text{QS}^2 + \text{RT}^2)}$ is $\frac{4}{5}$.
Hence, the correct answer is $\frac{4}{5}$.