Trapezoid Midsegment Theorem
Lesson · Beginner
Geometry: Plane Geometry
MN is the midsegment of trapezoid ABCD.
Assume, for contradiction, that MN ∦ AB.
Draw parallel lines from M and N.
By Triangle Proportionality Theorem in triangle DAB,
Therefore, E is the midpoint of DB.
By Triangle Proportionality Theorem in triangle DBC,
Therefore, E' is the midpoint of DB.
By Triangle Proportionality Theorem in triangle DAB,
Similarly,
Prove that
Similarly,
In trapezoid ABCD, AB ∥ CD, with AB < CD. Points M and N are the midpoints of AD and BC.
The areas of trapezoids ABNM and MNCD are in the ratio
If MN = 12, find AB and CD.
Let AB = x and CD = y.
By (1), (2),
Let G be the centroid of triangle ABC, and let ℓ be an arbitrary line through G. Perpendiculars from A, B, and C to ℓ have lengths x, y, and z, respectively, as shown. Prove that x = y + z.
Draw median AM and a perpendicular from M to ℓ.
By the Trapezoid Midsegment Theorem,
so
By AA,
By (1), (2),
Since AD = x,