2004 AIME I Problem 9

Let A B C be a triangle with sides 3,4 , and 5 , and D E F G be a 6 -by- 7 rectangle. A segment is drawn to divide triangle A B C into a triangle U_{1} and a trapezoid V_{1}, and another segment is drawn to divide rectangle D E F G into a triangle U_{2} and a trapezoid V_{2} such that U_{1} is similar to U_{2} and V_{1} is similar to V_{2}. The minimum value of the area of U_{1} can be written in the form m / n, where m and n are relatively prime positive integers. Find m+n.