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If $CE:EA:AD:DB=2:3:4:5,$ then what is $[ACD]/[ABE]?$

 Jan 27, 2021
 #1
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[ACD] / [ABD] = 20 / 27

 Jan 27, 2021
 #2
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Thank you! :D

 

Here's the system's explanation:

 

We start by converting part-to-part ratios into part-to-whole ones. We know that $CE:EA=2:3$ and $AD:DB=4:5$, so $AE:AC=3:5$ and $AD:AB=4:9$.

 

Consider $\triangle ABC$. On the one hand, $\triangle ACD$ and $\triangle ABC$ share a height, so $\dfrac{[ACD]}{[ABC]}=\dfrac{AD}{AB}=\dfrac{4}{9}$. On the other hand, $\triangle ABE$ and $\triangle ABC$ share a height, so $\dfrac{[ABE]}{[ABC]}=\dfrac{AE}{AC}=\dfrac{3}{5}$.

Therefore, $\dfrac{[ACD]}{[ABE]}=\dfrac{(4/9)[ABC]}{(3/5)[ABC]}=\dfrac{4\cdot 5}{9\cdot 3}=\boxed{\dfrac{20}{27}}$.

Guest Jan 27, 2021
 #3
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+2

[ACD] = 1/2 (AC * AD) = 20

 

[ABE] = 1/2 (AE * AB) = 27

 

jugoslav  Jan 27, 2021

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