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?
geometry
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?
Best Answer
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:
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.