Problem:
Equilateral â–³ABC has side length 1, and squares ABDE,BCHI, and CAFG lie outside the triangle. What is the area of hexagon DEFGHI?
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Answer Choices:
A. 412+33​​
B. 29​
C. 3+3​
D. 26+33​​
E. 6
Solution:
The three squares each have area 1, and △ABC has area 43​​. Note that ∠EAF=360∘−60∘−2⋅90∘=120∘. Thus the altitude from A in isosceles △EAF partitions the triangle into two 30−60−90∘ right triangles, each with hypotenuse 1. It follows that △EAF has base EF=3​ and altitude 21​, so its area is 43​​. Similarly, triangles GCH and DBI each have area 43​​. Therefore the area of hexagon DEFGHI is 3⋅43​​+3⋅1+43​​=(C)3+3​​.
