Competition problem — Geometry — Maths Olympiad Prep

Library / /183 of 520

Geometry Difficulty 5.0 AIME, harder Find the answer

8. In Rt $\triangle A B C$ , the altitude $C D$ on the hypotenuse $A B$ is $3$ , extend $D C$ to point $P$ , such that $C P=2$ , connect $A P$ , draw $B F \perp A P$ , intersecting $C D$ and $A P$ at points $E$ and $F$ respectively. Then the length of segment $D E$ is $\qquad$

Your answer

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

Solution

8. $\frac{9}{5}$ .
As shown in Figure 5, let $A D=a$ .
In the right triangle $\triangle A B C$ , given $C D=3, C D \perp A B$ , we know $B D=\frac{9}{a}$ .
Notice that, in the right triangles $\triangle A P D \sim \triangle E B D$ ,

then $\frac{D E}{B D}=\frac{A D}{P D} \Rightarrow D E$ $=\frac{9}{a} \times \frac{a}{5}=\frac{9}{5}$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

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