Chapter 13, Problem 17CUR

To determine

To prove: The median of a trapezoid is parallel to each base.

Explanation of Solution

Given information: A quadrilateral is trapezoid and its median.

Proof:

Consider a trapezoid OABC with the coordinates as shown below.

  

It is known that the two sides of trapezoid are parallel and two sides are nonparallel. Form the figure it can be seen that OA is parallel to CB.

Also, P is the midpoint of OC and Q is the midpoint of AB.

By the midpoint formula, the coordinates of point P are:

  $\left(\frac{0+a}{2},\frac{0+d}{2}\right)=\left(\frac{a}{2},\frac{d}{2}\right)$

By the midpoint formula, the coordinates of point Q are:

  $\left(\frac{a+b+c+a+b}{2},\frac{0+d}{2}\right)=\left(\frac{2a+2b+c}{2},\frac{d}{2}\right)$

As the line OA is on x -axis, so its slope is zero. Since, $\text{OA}\parallel \text{BC}$ , so the slope of the line BC is 0.

Now, calculate the slope of PQ as:

  $\begin{array}{c}m=\frac{\frac{d}{2}-\frac{d}{2}}{\frac{2a+2b+c}{2}-\frac{a}{2}}\\ =\frac{0}{\frac{2a+2b+c}{2}-\frac{a}{2}}\\ =0\end{array}$

As the slope of OA, CB and PQ are the same and zero, so the median PQ is parallel to each base of the trapezoid.

Hence, median of a trapezoid is parallel to each base is proved.