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The diagonals of a trapezoid are perpendicular and have lengths 8 and 10. Find the length of the median of the trapezoid.

It this possible without a rhombus?

Bob Joe
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2 Answers2

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Let $ABCD$ a trapezoid such that $FG$ is its median, $AC$ and $DB$ are perpendicular, $DC=w$, $AB =z$, $AC=8$ and $DB=10$.

Let $r$ such that $r \parallel AC$ and $D \in r$.

Let $s$ such that $A \in s$ and $B \in s$.

Let point $E$ such that $\{E\} =r \cap s$. See the figure below:

enter image description here

It follows that: $$DE=CA=8,$$ $$EA=w,$$ $$FG = \frac{w+z}{2} \quad (1)$$ and $$DE \perp DB.$$ Using the Pythagorean Theorem in $\triangle EDB$, we get: $$w+z= \sqrt{164} = 2 \sqrt{41}.$$ From $(1)$ we get: $$FG= \sqrt{41}.$$ Therefore it is possible to determine $FG$ without assuming a rhombus.

RicardoCruz
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Trapezoid can never be a rhombus - or more generally a parallelogram - because a trapezoid, by definition, can only have one set of parallel sides (also called bases).

RohitK
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    While you may have a point about trapezoids being distinguished by definition from rhombi, there is still a problem in the Question that you did not address. Answering a two year old Question should emphasize thoroughness in presentation rather than brevity. – hardmath Feb 01 '17 at 05:39