Problem:
Equiangular hexagon ABCDEF has side lengths AB=CD=EF=1 and BC=DE=FA=r. The area of â–³ACE is 70% of the area of the hexagon. What is the sum of all possible values of r?
Answer Choices:
A. 343​​
B. 310​
C. 4
D. 417​
E. 6
Solution:
Triangles ABC,CDE and EFA are congruent, so △ACE is equilateral. Let X be the intersection of the lines AB and EF and define Y and Z similarly as shown in the figure. Because ABCDEF is equiangular, it follows that ∠XAF=∠AFX=60∘. Thus △XAF is equilateral. Let H be the midpoint of XF. By the Pythagorean Theorem,
