Question

How do you prove this theorem on trapezoids and it’s medians? The median (or mid segment) of a trapezoid is parallel to each base and it’s length is one half the sum of the lengths of the bases.

Answer 1

Hint: Construct a trapezoid and name the midpoints of the side legs. Extend lower base such that it meets the segment passing from the first leg and the midpoint of its opposite leg. Try to prove that the midpoint of the second side leg is the midpoint of the segment passing through the end of the first leg and meeting the extended lower base.

Complete step-by-step answer:
Let ABCD be a trapezoid. Let AD and BC be the lower and upper base respectively. Let M be the midpoint of AB and N be the midpoint of CD.
Now connect the point B with N and extend the segment such that it meets the extended base AD at the point say X as shown in the figure.

Consider the two triangles $ \vartriangle BCN $ and $ \vartriangle NDX $ .
we will try to prove that these two triangles are congruent.
1. $ \angle BCN $ and $ \angle NDX $ are vertical
2. As N is the midpoint of CD, segments CN and Nd are congruent.
3. $ \angle BCN $ and $ \angle NDX $ are alternate interior angle triangles with parallel lines BC and DX and transversal CD.
Hence triangles $ \vartriangle BCN $ and $ \vartriangle NDX $ are congruent by angle side angle theorem.
Hence Segments BC and DX are congruent and BN and NX are congruent, which takes us to the conclusion that N is midpoint of BX.
Now consider $ \Delta ABX $ .
Since M is midpoint of AB and N is midpoint of BX, MN is the mid segment of $ \Delta ABX $ and hence parallel to AX and hence half of AX.
But AX is the sum of lower bases AD and DX , which is congruent to upper base BC.
Hence we can conclude that MN is equal to half of the sum of two bases AD and BC.

Note: Trapezoids are identified by the fact having two sides parallel and two sides non-parallel .The above kind of trapezoids are called as isosceles trapezoids where the non-parallel sidTare congruent. There are many properties of the isosceles trapezoid out of which we have proven one above.