14. The diagram shows a square $P Q R S$ of side-length 1 . $W$ is the centre of the square and $U$ is the midpoint of $R S$. Line segments $T W, U W$ and $V W$ split the square into three regions of equal area. What is the length of $S V$ ?
A $\frac{1}{2}$
B $\frac{2}{3}$
C $\frac{3}{4}$
D $\frac{4}{5}$
E $\frac{5}{6}$
Solution
E

Let the length of $S V$ be $x$. Since the three areas are equal, each must be equal to one third. We know $U W=\frac{1}{2}$ and $S U=\frac{1}{2}$, so the area of the trapezium $S V W U$ is $\frac{1}{2} \times\left(x+\frac{1}{2}\right) \times \frac{1}{2}=\frac{1}{3}$. Multiplying both sides by 4 , we get $\left(x+\frac{1}{2}\right)=\frac{4}{3}$ so $x=\frac{4}{3}-\frac{1}{2}=\frac{5}{6}$.