Enough Information?
The diagram below shows a smaller pentagon formed inside a large, convex pentagon by joining each vertex of the large pentagon with its non-adjacent vertices. Each of the 5 colored triangles has the same area of 1. Find the ratio of the area of the large pentagon to the area of the smaller pentagon.
If the ratio can be expressed as where are positive integers, is square-free, input the smallest possible value of as your answer.
Diagram not drawn to scale
The answer is 17.
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1 solution
The coloured triangles certainly all have the same area in a regular pentagon. For a regular pentagon of side a , the smaller pentagon formed by the intersection of its diagonals has side ϕ 2 a where ϕ is the golden ratio, so the ratio of areas is ϕ 4 = 2 7 + 3 5 giving the answer 1 2 .
This is enough to answer the question - but perhaps a bonus question would be to prove that the condition of the small triangles having equal area necessarily means we get this ratio of areas.