Mock Geometry AIME 2011 Problems/Problem 4
In triangle Let and be points on such that is equilateral. The perimeter of can be expressed in the form where are relatively prime positive integers. Find
Let be the midpoint of . It follows that is perpendicular to and to . The area of can then be calculated two different ways: , and .
By the Law of Cosines, and so . Therefore, . Solving for yields .
Let be the side length of . The height of an equilateral triangle is given by the formula . Then . Solving for yields . Then the perimeter of the triangle is and .
Let and . By the Law of Cosines, . It is easy to see that . Since , by AA similarity. From this, we have: Solving, we find that , so the perimeter is , and our answer is