# Lengths in a Crossed Trapezium +-- {.image} [[LengthsinaCrossedTrapezium.png:pic]] > In this right-angled trapezium, the green area is $6$ more than the yellow area. What's the length of the sloping side? =-- ## Solution by [[Similar Triangles]] and [[Pythagoras' Theorem]] +-- {.image} [[LengthsinaCrossedTrapeziumLabelled.png:pic]] =-- In the above diagram, let $a$ and $b$ be the lengths of the parallel sides in the trapezium, so that $G E$ has length $a$ and $A D$ has length $b$. Triangles $G H E$ and $A H B$ are [[similar]], so $F H$ and $H C$ are in the ratio $a : b$. Since $F C$ has length $3$, this means that $F H$ has length $\frac{3 a}{a + b}$ and $H C$ has length $\frac{3 b}{a + b}$. The area of triangle $G H C$ is $\frac{3 a^2}{2(a + b)}$ and of $A H D$ is $\frac{3 b^2}{2(a + b)}$ so: $$ 6 = \frac{3 b^2}{2(a + b)} - \frac{3 a^2}{2 (a + b)} = \frac{3(b^2 - a^2)}{2(a + b)} = \frac{3}{2} (b - a) $$ Therefore $b - a = 4$. Applying [[Pythagoras' theorem]] to triangle $G B A$ then shows that $$ x^2 = 4^2 + 3^2 = 25 $$ and so $x = 5$.