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All of the triangles are right-angled with integer sides. If AB = 3, what is the length of AE?

 

 Jul 27, 2020
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Since all sides are integers, we can use pythagorean tripples to help solve this. If you dont know what that is, you can look here:

https://www.mathsisfun.com/pythagorean_triples.html (in general, mathisfun.com is a great place to search up the definition of a math term, as they give many cool diagrams and their wording is simple and easy to understand)

 

Anyways, we see that since BA is 3, and all sides are integers, CB has to be 4, and CA has to be 5 (3-4-5 triangle). From here, we see that since CA is the shorter leg of the blue triangle, CD has to be 12, and DA has to be 13 (5-12-13 triangle). And finally, the green triangle. Now, although DA looks like the longer leg, the only other pythagorean triple that contains 13 is a 13-84-85 triangle. So, here is a great example on why you should always use geometric intuition, but never assume. Therefore, the length of AE is 85.

 Jul 27, 2020

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