2004 AIME I Problems/Problem 9
Problem
Let be a triangle with sides 3, 4, and 5, and be a 6by7 rectangle. A segment is drawn to divide triangle into a triangle and a trapezoid and another segment is drawn to divide rectangle into a triangle and a trapezoid such that is similar to and is similar to The minimum value of the area of can be written in the form where and are relatively prime positive integers. Find
Solution
We let for the purpose of labeling. Clearly, the dividing segment in must go through one of its vertices, without loss of generality . The other endpoint () of the segment can either lie on or . is a trapezoid with a right angle then, from which it follows that contains one of the right angles of , and so is similar to . Thus , and hence , are s.
Suppose we find the ratio of the smaller base to the larger base for , which consequently is the same ratio for . By similar triangles, it follows that by the same ratio , and since the ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding lengths, it follows that .


 If lies on , then ; the latter can be discarded as extraneous. Therefore, , and the ratio . The area of in this case.
 If lies on , then ; the latter can be discarded as extraneous. Therefore, , and the ratio . The area of in this case.
Of the two cases, the second is smaller; the answer is , and .
See also
2004 AIME I (Problems • Answer Key • Resources)  
Preceded by Problem 8 
Followed by Problem 10  
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15  
All AIME Problems and Solutions 
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