## Trapzium Cut in Half By Line Parallel to the Parallel Sides

Suppose we have a trapezium with parallel sides 10 and 4, and height 4. The trapezium is to be cut in half by a line parallel to the parallel sides. Let the distance of this line from the longer parallel side be
$x$
.

We must find an equation in terms of
$x$
for the length
$l$
of the cutting line. When
$x=0, \: l=10$
and when
$x=4, \: l=4$
.
From these two,
$l=10-1.5x$

The area of the part above the cutting line is
$\frac{1}{2}(4+10-1.5x)(4-x)$

The area of the part below the cutting line is
$\frac{1}{2}(10+10-1.5x)x$

Equating these gives
$\frac{1}{2}(4+10-1.5x)(4-x)=\frac{1}{2}(10+10-1.5x)x$

Expanding this and simplifying gives
$3x^2-40x+56=0$

Solving this equation gives
$x= \frac{40 \pm \sqrt{928}}{6}$

Obviously x must be less than the height of the trapezium so
$x= \frac{40 - \sqrt{928}}{6}= x= \frac{20 - 2\sqrt{58}}{3}$
since the other possibility gives
$x \gt 4$
.