jmc

Question

An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is 23 the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle? FIGURE

Accepted Answer

All equilateral triangles are similar. For the outer equilateral triangle to the inner equilateral triangle, since their side-length ratio is 32, their area ratio is (32)^2=94. It follows that the area ratio of three trapezoids to the inner equilateral triangle is 94-1=54, so the area ratio of one trapezoid to the inner equilateral triangle is 5413=512=5 : 12.