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median of trapezoid
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(Theorem)
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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 $AB$ and $CD$ be the bases of a trapezoid $ABCD$ and $E$ the midpoint of the leg $AD$ and $F$ the midpoint of the leg $BC$ . Then the median $EF$ may be determined as vector as follows:
The last expression tells that $\overrightarrow{EF} \parallel \overrightarrow{AB}+\overrightarrow{DC} \parallel \overrightarrow{AB}$ and $\displaystyle|\overrightarrow{EF}| = \frac{|\overrightarrow{AB}\!+\!\overrightarrow{DC}|}{2} = \frac{|\overrightarrow{AB}|\!+\!|\overrightarrow{DC}|}{2}$ . Q.E.D.
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"median of trapezoid" is owned by pahio.
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Cross-references: expression, vector, leg, proof, arithmetic mean, length, bases, parallel, median, trapezoid, midpoints, segment
There is 1 reference to this entry.
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 2262 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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Pending Errata and Addenda
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