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imc

geometry intermediate

Problem

Two equilateral triangles are contained in square whose side length is . The bases of these triangles are the opposite side of the square, and their intersection is a rhombus. What is the area of the rhombus?
(A)
(B)
(C)
(D)
Solution
Observe that the rhombus is made up of two congruent equilateral triangles with side length equal to GF. Since AE has length and triangle AEF is a 30-60-90 triangle, it follows that EF has length 1. By symmetry, HG also has length 1. Thus GF has length . The formula for the area of an equilateral triangle of length is . It follows that the area of the rhombus is:
Final answer
D