2017 AMC 10A Problems/Problem 21
A square with side length is inscribed in a right triangle with sides of length , , and so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length is inscribed in another right triangle with sides of length , , and so that one side of the square lies on the hypotenuse of the triangle. What is ?
Analyze the first right triangle.
Note that and are similar, so . This can be written as . Solving, .
Now we analyze the second triangle.
Similary, and are similar, so , and . Thus, . Solving for , we get . Thus, .