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.
- Prove that E and F lie on the midsegment of the trapezoid
- If AB=10 and DC = 22, find EF
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.
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.
$PQ = \frac 12 (AB + CD)$