A question from an ACT Math test:
Please teach me a way to get the answer quickly, I must've missed some properties of trapezoid cause I can only think of making similar triangles to find the answer but I'll have to assume two unknown numbers, which is too time-consuming.
$\endgroup$1 Answer
$\begingroup$The Trapezoid Midsegment Theorem states that
$ABCD$ is a trapezoid with $AB||CD$. $EF$ is a line connecting the midpoints of sides $AD$ and $BC$, that is $AE=ED$ and $BF=FC$. Then, $EF||DC$ and that $EF=\frac{1}{2}(AB+DC)$.
For a proof of the theorem check Trapezoid Midsegment Theorem Proof: Geometry Help.
Use this to go through the following solution:$$\overline{EF}=\frac{\overline{BC}+\overline{AD}}{2}\Rightarrow \overline{AD}=38\mathrm{~inches},\\\overline{GH}=\frac{\overline{EF}+\overline{AD}}{2}=31.5\mathrm{~inches}.$$
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