🔥 New! 2025 AMC 10 B

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Problem 13 Easy

The altitude to the hypotenuse of a 30-60-9 0 ^ { \circ } right triangle is divided into two segment of lengths x< y by the median to the shortest side of the triangle. What is the ratio \frac {x}{x+y}?

  • A.

    \frac { 3 } { 7 }

  • B.

    \frac { \sqrt { 3 } } { 4 }

  • C.

    \frac { 4 } { 9 }

  • D.

    \frac { 5 } { 1 1 }

  • E.

    \frac { 4 \sqrt { 3 } } { 1 5 }

Answer:A
Solution

ith Menelaus Theorem on \triangle BCE:

\frac{EF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CA}{AE} = 1

Since BD = CD:

\frac{AC}{AE} = \frac{AC}{AB} \cdot \frac{AB}{AE} = \frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{3}} = \frac{4}{3}

\frac{EF}{FB} = \frac{AE}{AC} = \frac{3}{4} Thus \frac{x}{x+y} = \frac{3}{3+4} = \frac{2}{7}

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