14. The diagram shows a rectangle $P Q R S$ in which $P Q: Q R=1: 2$. The point $T$ on $P R$ is such that $S T$ is perpendicular to $P R$.
What is the ratio of the area of the triangle $R S T$ to the area of the rectangle $P Q R S$ ?
A $1: 4 \sqrt{2}$
B $1: 6$
C $1: 8$
D $1: 10$
Solution
D We can suppose that we have chosen units so that the length of $R S$ is 1 . So the length of $P S$ is 2. It follows from Pythagoras' theorem that the length of $P R$ is $\sqrt{1^{2}+2^{2}}$, that is, $\sqrt{5}$.
The triangles $R S T$ and $P R S$ are both right-angled and the angle at $R$ is common to both triangles. Therefore the triangles are similar. Hence the ratio of their areas is the ratio of the squares of the lengths of corresponding sides. The lengths of their hypotenuses are in the ratio $1: \sqrt{5}$. Hence, the ratio of the area of triangle $R S T$ to the area of triangle $P R S$ is $1^{2}: \sqrt{5}^{2}$, that is, $1: 5$. The area of triangle $P R S$ is half the area of the rectangle. Hence the ratio of the area of triangle $R S T$ to the area of the rectangle is $1: 10$.