Question

In trapezoid $EFGH,$ $\overline{EF} \parallel \overline{GH}$. Let $P$ be the midpoint of $\overline{EH}$. If the area of triangle $EFP$ is $8$, and the area of triangle $GHP$ is $21$, then find the area of trapezoid $EFGH$.

Answer 1

72

Explanation:

I took the test

Answer 2

58

Explanation:

Given that in trapezoid EFGH, (picture enclosed)

EF is parallel to GH.

P is the mid point of non parallel side EH, let Q be mid point of non parallel side FG

Since EP = PH and FQ = QG

by parallel lines when make equal intercepts we get

PQ = 1/2 (EF+FG)

Area of trapezium = 1/2 (EF+FG)*h where h is the altitude

We find that height for both triangles EFP and GHP would be the same from equal intercepts theorem for any transversal.

So we get 1/2 * EF*h/2 = 8 and 1/2 *GH *h/2 = 21

i.e. (EF+FG)h/4 =29

So area = 1/2 (EF+FG)h = 58