+1347 Let $A$, $B$, $C$, $D$, and $E$ be points in the plane such that:
* $D$ is on line $AE$ with $\overline{CD} \perp \overline{AE}$.
* $B$ is on line $CE$ with $\overline{AB} \perp \overline{CE}$.
* $AB = 4$, $BE = 3$, and $CE = 8$.
Find the length $DE$.
+128707 C
5
B
4 3
A D E
AE = sqrt [ AB^2 + BE^2] = sqrt [ 4^2 + 3^2 ] = sqrt [25] = 5
AC = sqrt [ BC^2 + AB^2] = sqrt [ 5^2 + 4^2] = sqrt [41]
Let DE = x and AD = 5 - x
CD = sqrt [ (sqrt 41)^2 - (5 - x)^2 ]
CD = sqrt [ 8^2 - x^2 ]
Equate these
sqrt [ (sqrt 41)^2 - (5 -x)^2] = sqrt [ 8^2 - x^2 ] square both sides
(sqrt 41)^2 - (5 -x)^2 = 8^2 - x^2
41 - x^2 + 10x - 25 = 64 - x^2
10x + 16 = 64
10x = 48
x = 48/10 = 4.8 = DE
