PlanetMath: median of trapezoid

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median of trapezoid

(Theorem)

The segment connecting the midpoints of the legs of a trapezoid, i.e. the median of the trapezoid, is parallel to the bases and its length equals the arithmetic mean of the legs.

Proof. Let and be the bases of a trapezoid and the midpoint of the leg and the midpoint of the leg . Then the median may be determined as vector as follows:

$\displaystyle \overrightarrow{EF}$	$\displaystyle = \overrightarrow{ED}+\overrightarrow{DC}+\overrightarrow{CF}$
	$\displaystyle = \frac{1}{2}\overrightarrow{AD}+\overrightarrow{DC}+\frac{1}{2}\overrightarrow{CB}$
	$\displaystyle = \frac{1}{2}(\overrightarrow{AD}+\overrightarrow{DC}+\overrightarrow{CB})+\frac{1}{2}\overrightarrow{DC}$
	$\displaystyle = \frac{1}{2}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{DC}$
	$\displaystyle = \frac{1}{2}(\overrightarrow{AB}+\overrightarrow{DC})$

The last expression tells that $\overrightarrow{EF} \parallel \overrightarrow{AB}+\overrightarrow{DC} \parallel \overrightarrow{AB}$ and $\displaystyle\vert\overrightarrow{EF}\vert = \frac{\vert\overrightarrow{AB}\!+... ...{2} = \frac{\vert\overrightarrow{AB}\vert\!+\!\vert\overrightarrow{DC}\vert}{2}$ . Q.E.D.

$\begin{pspicture}(-1,-1)(7,4) \psline[linecolor=blue](0,0)(6,0) \psline(6,0)(5,4... ...rput[a](1.74,3){$p$} \rput[a](5.55,1){$q$} \rput[a](5.05,3){$q$} \end{pspicture}$

"median of trapezoid" is owned by pahio.

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Cross-references: expression, vector, leg, arithmetic mean, length, bases, parallel, median, trapezoid, midpoints, segment
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This is version 3 of median of trapezoid, born on 2008-02-05, modified 2008-05-23.
Object id is 10237, canonical name is MedianOfTrapezoid.
Accessed 563 times total.

Classification:

AMS MSC:	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	51M25 (Geometry :: Real and complex geometry :: Length, area and volume)

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