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smc

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. The ratio of the area of the other small right triangle to the area of the square is
(A)
(B)
(C)
(D)
Solution
WLOG, let a side of the square be . Simple angle chasing shows that the two right triangles are similar. Thus the ratio of the sides of the triangles are the same. Since , the base of the triangle with area is . Therefore where is the height of the other triangle. , and the area of that triangle is .
Final answer
D