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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 $DE$.

 Sep 7, 2023
edited by sandwich  Sep 7, 2023
 #1
avatar+129690 
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              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

 

 

cool cool cool

 Sep 7, 2023

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