+380 In trapezoid EFGH, \overline{EF} \parallel \overline{GH}, and P is the point on \overline{EH} such that EP:PH = 1:2. If the area of triangle PEF is 6, and the area of triangle PGH is 6, then find the area of trapezoid EFGH.
+130068 E F
M P
N H G
Let EN be the height
Triangle ENH is right
Since EP : PH = 1 : 2
Then EN = 1 / ( 1 + 2) * height = 1/3 height of trapezoid
So height of triangle PGH = 2/3 height of trapezoid
[ EPF] = (1/2)EF * (1/3) height → EF * height / 6 = 6 → EF = 36 / height
[ PGH ] = (1/2)GH * (2/3) height = EF * height / 3 = 6 → GH = 18 / height
[ EFGH ] = (1/2) height * [ 36 / height + 18 / height ] = (1/2) height [ 54 / height ] =
54 / 2 = 27
