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# help geometry

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In trapezoid $EFGH,$ $\overline{EF} \parallel \overline{GH},$ and $P$ is the midpoint of side $\overline{EH}$. If the area of triangle $PEF$ is $18$, and the area of triangle $PGH$ is $36$, then find the area of trapezoid $EFGH$.

Jun 25, 2024

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To find the area of trapezoid EFGH, we can use the fact that the area of a trapezoid is equal to the average of the lengths of the two bases multiplied by the height. Given that P is the midpoint of side EH, we can use the areas of triangles PEF and PGH to find the height of trapezoid EFGH. 1. Since P is the midpoint of EH, we know that the height of trapezoid EFGH is equal to the distance between EF and GH. 2. The area of triangle PEF is given as $18.Let′sdenotethebaseoftrianglePEFasx.Therefore,theheightoftrianglePEFisalsoh,wherehisthedistancebetween\overline{EF}and\overline{GH}.3.Usingtheformulafortheareaofatriangle,wehave\frac{1}{2} \times x \times h = 18.Thus,x \times h = 36.4.Similarly,theareaoftrianglePGHisgivenas36, and let's denote the base of triangle PGH as y. Therefore, y×h=72. 5. Now, the average of the bases of trapezoid EFGH is 21​(x+y). Multiplying this by the height h, we get the area of trapezoid EFGH as 21​(x+y)×h=21​(xh+yh)=21​(36+72)=54. Therefore, the area of trapezoid EFGH is$54\$ square units.