2012 AMC 10B Problems/Problem 21
Four distinct points are arranged on a plane so that the segments connecting them have lengths , , , , , and . What is the ratio of to ?
When you see that there are lengths a and 2a, one could think of 30-60-90 triangles. Since all of the other's lengths are a, you could think that . Drawing the points out, it is possible to have a diagram where . It turns out that and could be the lengths of a 30-60-90 triangle, and the other 3 can be the lengths of an equilateral triangle formed from connecting the dots. So, , so