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In triangle $ABE,$ $C$ and $D$ are points on sides $\overline{BE}.$ If $BD = 8$, $CE = 24$, $[ABC] = 4$, and $[ADE] = 8$, then find $[ACD]$.

 Jun 5, 2024
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                     A

 

        

     

B  8     D      x         C     24             E

 

[ABC] = 4

 

So

4 = (1/2) (8 + x) (h)

8 / (8 + x)  = h    (1)

 

And   

 

[ADE ] = 9

 

So

8 =(1/2) (24 + x) h

16 / (24 +x)  = h   (2)

 

Equate (1) (2)

 

8 /(8 + x)  = 16 / (24 + x)

 

1/ (8 + x)  = 2 /(24 + x)

 

2 (8 + x) = 24 + x

 

16 + 2x  = 24 + x

 

x = 8

 

h = 8 / ( 8 + 8)    = 1/2

 

So

 

[ ACD ]  = (1/2) ( 8) (1/2)  =  2

 

cool cool cool

 Jun 5, 2024

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