imc

Question

Square ABCD has side length 1. Points P, Q, R, and S each lie on a side of ABCD such that APQCRS is an equilateral convex hexagon with side length s. What is s?

Accepted Answer

Note that BP=BQ=DR=DS=1-s. It follows that BPQ and DRS are congruent isosceles right triangles. In BPQ, we have PQ=BP2, or align* s &= (1-s)2 s &= 2 - s2 (2+1)s &= 2 s &= 22 + 1. align* Therefore, the answer is s = 22 + 12 - 12 - 1 = 2 - 2.