Track / Stage 4 / 69 of 340 #329 of 2000

Problem 329

AMC 12 late, AIME early

Geometry Difficulty 4.6 Find the answer

Example 4 As shown in Figure 4, in rhombus $A B C D$ , $\angle A B C$ $=120^{\circ}, F$ is the midpoint of $D C$ , and the extension of $A F$ intersects the extension of $B C$ at point $E$ . Then the degree measure of the acute angle formed by line $B F$ and $D E$ is ( ).
(A) $30^{\circ}$
(B) $40^{\circ}$
(C) $50^{\circ}$
(D) $60^{\circ}$
(2008, National Junior High School Mathematics League Wuhan CASIO Cup Selection Competition)

Your answer

Official solution

As shown in Figure 4, let $B F$ intersect $D E$ at point $M$ , and connect $B D$ . Then $\triangle B C D$ is an equilateral triangle.
From $F$ being the midpoint of $C D$ , we know $\angle M B C=30^{\circ}$ .
Since $A D / / C E$ , then $\triangle A D F$ and $\triangle E C F$ are centrally symmetric about point
$F$ . Therefore, $C E=A D=C D$ .
Thus, $\angle C E M=30^{\circ}, \angle D M F=60^{\circ}$ .

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