Track / Stage 4 / 301 of 340 #561 of 1964

Problem 561

AMC 12 late, AIME early

Geometry Difficulty 5.0 Find the answer

3. In $\triangle A B C$ , it is known that $\angle A=a, C D, B E$ are the altitudes on $A B, A C$ respectively, then $\frac{D E}{B C}=$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

$|\cos a|$
3. 【Analysis and Solution】 $\because \angle B D C=\angle B E C, \therefore B, D, E, C$ are concyclic, $\angle A D E=\angle A C B, \triangle A E D \sim \triangle A B C$ , $\frac{D E^{2}}{B C^{2}}=\frac{S_{\triangle A E D}}{S_{\triangle A B C}}=\frac{A D \cdot A E}{A B \cdot A C}=\cos ^{2} a . \therefore \frac{D E}{B C}=|\cos a|$ .

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