[Math] Trapezoid midsegment diagonal proof

euclidean-geometrygeometry

Given trapezoid ABCD with bases AB and CD, draw diagonals AC and BD. Let E be the midpoint of AC and F the midpoint of BD.

  1. Prove that E and F lie on the midsegment of the trapezoid
  2. If AB=10 and DC = 22, find EF

Best Answer

let $P$ be the midpoint of $AD$ ans $Q$ be the midpoint of $BC$

$PQ$ is the midline of $ABCD$

The midline of a trapezoid is parallel to both bases.

$PE$ is the midline of $DAC$

$QF$ is the midline of $DBC$

The midline of a triangle is parallel to its base.

$PQ, PE, QF$ are all parallel to $CD$

By the parallel postulate, there exists a unique line is parallel to CD and passes through $P.$

$E,F$ and $Q$ lie on that line.

$PQ = \frac 12 (AB + CD)$

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