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jmc

geometry senior

Problem

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is times the area of the square. What is the ratio of the area of the other small right triangle to the area of the square? Express your answer as a common fraction in terms of .
Solution
Without loss of generality, let the side of the square have length 1 unit and let the area of triangle be . Let and . Because triangles and are similar, . Since , the area of triangle is .
Final answer
\frac{1}{4m}