Proving that the median of a trapezoid is parallel to the two bases.

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I'm asked to prove that the median of a trapezoid is parallel to the bases, and I'm told to prove this fact by showing that the midpoints of each leg of the trapezoid (i.e the two end points of the median line) are equidistant from one of the bases. I understand how proving this would show that the median line is parallel to the bases, but I don't know how to prove that the two distances are equal to each other.

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1 Answer

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Let $ABCD$ be our trapezoid, $BC||AD$, $M$ and $N$ be mid-points of $AB$ and $CD$ respectively.

Also, let $K$ be a mid-point of $AC$.

Thus, $MK||BC$ and $KN||AB$, which since $AB||BC,$ gives $MK||BC$ and $KN||BC,$ which says us that $M$, $K$ and $N$ are placed on the same line and we got $MN||BC.$

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