16. The diagram shows two right-angled triangles inside a square. The perpendicular edges of the larger triangle have lengths 15 and 20 . What is the area of the shaded quadrilateral?
A 142
B 146
C 150
D 154
E 158
Solution
D
We let the vertices of the square be $P, Q, R$ and $S$, and the points $T$ and $U$ be as shown.

We note first that, by Pythagoras' Theorem, applied to the right-angled triangle $T Q R$,
$$
R T^{2}=15^{2}+20^{2}=225+400=625=25^{2} .
$$

It follows that $R T=25$.
Because $S R$ is parallel to $P Q$, the alternate angles, $\angle S R U$ and $\angle Q T U$ are equal. Therefore, the right-angled triangles $S U R$ and $R Q T$ are similar. Therefore their corresponding sides are in proportion. Hence
$$
\frac{S R}{R T}=\frac{S U}{R Q}=\frac{R U}{T Q} .
$$

Now $S R=R Q=20$. Hence
$$
\frac{20}{25}=\frac{S U}{20}=\frac{R U}{15} .
$$

It follows that $S U=16$ and $R U=12$.
The area of the shaded quadrilateral is the area of the square $P Q R S$ less the areas of the triangles $S U R$ and $R Q T$. It follows that the area of the shaded quadrilateral is
$$
20^{2}-\frac{1}{2}(16 \times 12)-\frac{1}{2}(20 \times 15)=400-96-150=154 .
$$