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Help! Geometry!

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In the diagram, BD=5, CE=4, [ABC]=24, and [ADE]=18. Find [ACD]

Feb 28, 2021

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The ratio of $\triangle ABC$ to $\triangle ACD$ to $\triangle ADE,$ which is just the ratio of the bases $BC$ to $CD$ to $DE$ (because the height for each of the triangles is the same). Therefore, we can write equations corresponding to these ratios. $$\frac{BC}{[ABC]} = \frac{DE}{[ADE]},$$  $$\frac{5-CD}{24} = \frac{4-CD}{18}.$$

Solving this set of equations, we find $$90-18CD = 96-24CD,$$ $$6CD=6,$$ $$CD = 1.$$

Therefore, we can use this value of $CD$ to substitute back into the original equation. $$\frac{CD}{[ACD]} = \frac{DE}{[ADE]},$$ $$\frac{1}{[ACD]} = \frac{3}{18},$$ $$[ACD] = 6.$$

Thus, our answer is $\boxed{[ACD] = 6}.$

(Note: The diagram is not drawn to scale).

Feb 28, 2021