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1. The diagram below shows twelve 30-60-90 triangles placed in a circle so that the hypotenuse of each triangle coincides with the longer leg of the next triangle. The fourth and last triangle in this diagram are shaded. The ratio of the perimeters of these two triangles can be written as m/n where m and n are relatively prime positive integers. Find m+n.

 

2. The midpoints of a regular hexagon are connected to form a smaller hexagon. The small hexagon has perimeter 2√3. What is the perimeter of the large hexagon?

 Dec 15, 2019
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1. The ratio of the perimeters is (2/sqrt(3))^4 = 16/9, so the answer is 16 + 9 = 25.

 

2. The perimeter of the large hexagon is 6/sqrt(3)*2*sqrt(3) = 12.

 Dec 15, 2019

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