15. The diagram shows a square $P Q R S$. Points $T, U, V$ and $W$ lie on the edges of the square, as shown, such that $P T=1$, $Q U=2, R V=3$ and $S W=4$.
The area of $T U V W$ is half that of $P Q R S$.
What is the length of $P Q$ ?
A 5
B 6
C 7
D 8
E 9
Solution
B
We let the side length of the square $P Q R S$ be $x$. Then the lengths of $T Q, U R, V S$ and $W P$ are $x-1, x-2, x-3$ and $x-4$, respectively.
The area of the square $P Q R S$ is $x^{2}$. The area of $T U V W$ is a half of this. It follows that the sum of the areas of the triangles $P T W, T Q U, U R V$ and $V S W$ is also a half of the area of the square, and hence this sum equals $\frac{1}{2} x^{2}$.
The area of a triangle is half the product of its base and its height.
We therefore have
$$
\frac{1}{2}(1 \times(x-4))+\frac{1}{2}(2 \times(x-1))+\frac{1}{2}(3 \times(x-2))+\frac{1}{2}(4 \times(x-3))=\frac{1}{2} x^{2} .
$$

This equation simplifies to give
$$
10 x-24=x^{2}
$$
and therefore
$$
x^{2}-10 x+24=0 .
$$

The left-hand side of the last equation factorizes to give
$$
(x-4)(x-6)=0 .
$$

Hence $x=6$ or $x=4$.
For there to be four triangles, as shown in the diagram, $x>4$. We therefore deduce that $x=6$. So the length of $P Q$ is 6 .