Track / Stage 4 / 181 of 340 #441 of 1964

Problem 441

AMC 12 late, AIME early

Geometry Difficulty 4.8 Find the answer

81 Given a rhombus $A B C D$ with side length $a$ and $\angle A=\frac{\pi}{3}$ , the rhombus $A B C D$ is folded along the diagonal to form a dihedral angle $\theta$ , where $\theta \in\left[\frac{\pi}{3}, \frac{2 \pi}{3}\right]$ . Then the maximum distance between the two diagonals is
A. $\frac{3}{2} a$
B. $\frac{\sqrt{3}}{4} a$
C. $\frac{\sqrt{3}}{2} a$
D. $\frac{3}{4} a$

Your answer

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

Official solution

81 D. If folded along $B D$ , let the midpoint of $A C$ be $F$ , and the midpoint of $B D$ be $E$ , then $A E=\frac{\sqrt{3}}{2} a$ ,
$E F=A E \cos \frac{\theta}{2} \leqslant \frac{\sqrt{3}}{2} a \cos \frac{\pi}{6}=\frac{3}{4} a .$

If folded along $A C$ , let the midpoint of $A C$ be $F$ , and the midpoint of $B D$ be $E$ , then $B D=\frac{1}{2} a, E F=B D \cos \frac{\theta}{2} \leqslant$
$\frac{\sqrt{3} a}{4} .$

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