2025 AMC 12B Problem 10

Problem 10 of 25EasierGeometry

The altitude to the hypotenuse of a

30\text{-}60\text{-}90

right triangle is divided into two segments of lengths

x \lt y

by the median to the shortest side of the triangle. What is the ratio

\dfrac{x}{x+y}?

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Solution

Take

C = (0,0),

A = (\sqrt{3}, 0),

B = (0, 1),

AB

is the hypotenuse and

BC

is the shortest side. The altitude from

C

meets

AB

H = \left(\tfrac{\sqrt{3}}{4}, \tfrac{3}{4}\right).

The median from

A

M = \left(0, \tfrac{1}{2}\right)

crosses the altitude at

P = \left(\tfrac{\sqrt{3}}{7}, \tfrac{3}{7}\right).

This splits the altitude into

CP = \tfrac{2\sqrt{3}}{7} = \tfrac{4\sqrt{3}}{14}

and

PH = \tfrac{3\sqrt{3}}{14},

x = \tfrac{3\sqrt{3}}{14}

and

\dfrac{x}{x+y} = \dfrac{3}{7}.

Thus, the correct answer is A.