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In the diagram below, △ABC and △CDE are two right-angled triangles with AC = 24, CE =7 and ∠ ACB = ∠ CED. Find the length of the line segment AE.

Diagram:

The above is the diagram.

I came across this question in a Math Olympiad Competition. I am able to find out that △ABC and △CDE are similar triangles but after that, I am not sure what to do to solve the question. Can anyone help me with the solution? Thanks.

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4  
What can you say about $\angle ACE$? –  Blue Jun 4 at 7:30
1  
It is equal to 90 degrees? –  snivysteel Jun 4 at 7:33
    
@snivysteel Yes. –  dragon Jun 4 at 7:33
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Now use Pythagorean theorem: $$7^2+24^2=(\bar{AE})^2$$ and solve for $\bar{AE}$. –  dragon Jun 4 at 7:34
    
Oh, I did not realise this question was so simple. Thanks ! =D –  snivysteel Jun 4 at 7:36

1 Answer 1

up vote 3 down vote accepted

You don't even have to bother with similarity here (yes they are similar, but it doesn't matter).

Let $\angle ACB = \angle CED = \theta$. That means that $\angle ECD = 90^{\circ} - \theta$ by the angle sum of $\triangle CDE$.

That means that $\triangle ACE$ is a right triangle allowing to to apply Pythagoras' Theorem to it. So $AE = \sqrt{AC^2 + CE^2}=25$.

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