16. The figure shows two touching semicircles of radius 1 , with parallel diameters $P Q$ and $R S$. What is the square of the distance $P S$ ?
A 16
B $8+4 \sqrt{3}$
C 12
D 9
E $5+2 \sqrt{3}$
SOLUTION
B

Let $T$ and $U$ be the centres of the two semicircles and $V, W$ be where the perpendiculars from $U$ and $S$ meet $P Q$. Then $T U$ has length 2, and $U V$ and $S W$ have length 1 . Using Pythagoras' theorem $T V$ is $\sqrt{3}$.
This means that $P W$ is $2+\sqrt{3}$ as it is $T V+2$ radii. So $S P^{2}=(2+\sqrt{3})^{2}+1^{2}$ which is $8+4 \sqrt{3}$.