21. The diagram shows a right-angled triangle $P Q R$. The point $S$ is the midpoint of the side $Q R$ and $\tan \angle Q P R=\frac{3}{2}$.
What is the value of $\sin \angle Q P S$ ?
A $\frac{1}{\sqrt{3}}$
B $\frac{1}{\sqrt{2}}$
C $\frac{1}{2}$
D $\frac{3}{5}$
E $\frac{4}{5}$
21. D Let $P Q$ have length $4 x$. Then, as $\tan \angle Q P R=\frac{3}{2}, Q R$ has length 6x. Therefore, as $S$ is the midpoint of side $Q R, Q S$ has length $3 x$. By Pythagoras' Theorem in triangle $P Q S, P S^{2}=P Q^{2}+Q S^{2}=$ $(4 x)^{2}+(3 x)^{2}=16 x^{2}+9 x^{2}=25 x^{2}$. Hence $P S=\sqrt{25 x^{2}}=5 x$. So $\sin \angle Q P S=\frac{Q S}{P S}=\frac{3 x}{5 x}=\frac{3}{5}$.