## Median of a Trapezoid

The trapezoid $P\!ART$ has $m\angle P=x^\circ$ and $m\angle A=2x^\circ$. The three congruent sides have lengths $s$. Find the length of the median in terms of $s$.

Source: NCTM Mathematics Teacher 2008

SOLUTION

$m\angle P+m\angle A+m\angle R+m\angle T=360^\circ$
$x+2x+2x+x=360^\circ$
$6x=360^\circ$
$x=60^\circ$
Triangle $P\!AB$ is a $30^\circ-60^\circ-90^\circ$ triangle with hypotenuse equal $s$.
$P\!B=s/2$
Likewise $CT=s/2$
Median of trapezoid
$MN=\dfrac{AR+PT}{2}$
$=\dfrac{AR+PB+BC+CT}{2}$
$=\dfrac{s+s/2+s+s/2}{2}$
$=3s/2$

Answer: $3s/2$