Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME, harder Find the answer

$18 \cdot 110$ As shown, $A B C D$ is an isosceles trapezoid, $A D$ $=B C=5, A B=4, D C=10$ , point $C$ is on $D F$ , and $B$ is the midpoint of the hypotenuse of the right triangle $\triangle D E F$ . Then $C F$ equals
(A) 3.25 .
(B) 3.5 .
(C) 3.75 .
(D) 4.0 .
(E) 4.25 .

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

[Solution] Draw $B H \perp D F$ , then $B H / / E F$ . Since $A B C D$ is an isosceles trapezoid,
$\therefore \quad C H=\frac{1}{2}(D C-A B)=3,$

then
$D H=D C-C H=10-3=7 \text {. }$
Since $B$ is the midpoint of $D E$ , and $\frac{D B}{D E}=\frac{D H}{D F}$ ,
$\therefore D F=14 \text {. }$

Then $C F=D F-C D=4$ .
Therefore, the answer is $(D)$ .

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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.