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Problem 274

Geometry Difficulty 2.1 Find the answer CEMC Cayley

Points $A, B, C$ , and $D$ are on a line in that order. The distance from $A$ to $D$ is 24. The distance from $B$ to $D$ is 3 times the distance from $A$ to $B$ . Point $C$ is halfway between $B$ and $D$ . What is the distance from $A$ to $C$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Since $B$ is between $A$ and $D$ and $B D=3 A B$ , then $B$ splits $A D$ in the ratio $1: 3$ . Since $A D=24$ , then $A B=6$ and $B D=18$ . Since $C$ is halfway between $B$ and $D$ , then $B C=\frac{1}{2} B D=9$ . Thus, $A C=A B+B C=6+9=15$ .

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.