Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Find the answer

5. In trapezoid $ABCD$ , $AB \parallel CD, AB=3CD, E$ is the midpoint of diagonal $AC$ , and line $BE$ intersects $AD$ at $F$ . Then the value of $AF: FD$ is:

Pick one

Solution

5. C.

Draw $C G / / B F$ intersecting the extension of $A D$ at $G$ (as shown in the figure). From $A B$ $/ / C D, C G / / B F$ , we get $\triangle A B F \backsim$ $\triangle D C G$ , thus $A F: D G=A B$ ～ $D C=3: 1$ . Also, by $C G / / B F$ , $F D=\varepsilon D C$ , hence $A F: F l=-32=\frac{3}{2}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.