Checking if a Trapezium is Isosceles

An isosceles trapezium is a trapezium with a line of symmetry.
We are to decide if the trapezium ABCD below is isosceles. One way of doing this is to check whether the line between the midpoints of the parallel sides is perpendicular to these sides - the gradients multiply to give
$-1$
. If it is, the trapezium is isosceles.

The midpoint of BC is point P with coordinates
$\frac{(2,3)+(8,5)}{2}=(5,4)$

The midpoint of AD is point Q with coordinates
$\frac{(1,1)+(10,4)}{2}=(5.5,2.5)$

The gradient of QP is nbsp;
$\frac{(4-2.5}{5-5.5}=\frac{1.5}{-0.5}=-3$

The gradient of BC is nbsp;
$\frac{(5-3}{8-2}=\frac{2}{6}=\frac{1}{3}$

$-1$