+436 In triangle $ABE,$ $C$ and $D$ are points on sides $\overline{BE}.$ If $BD = 8$, $CE = 24$, $[ABC] = 4$, and $[ADE] = 8$, then find $[ACD]$.
+129771 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
