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The question given was to find the altitude to the side of length 14 in a triangle with sides of length 13,14,15. I keep trying proportions but I know I'm missing something or more likely, made a bad assumption somewhere. I did look up how to do it elsewhere which had the Pythagorean theorem twice and solving for a that way. What am I doing wrong here? I set up two similar triangles in my diagram to the right. Shouldn't the proportions match up and give the correct answer (a 5-12-13 and 9-12-15 triangle).

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    $\begingroup$ The two triangles are not similar $\endgroup$ – brainjam Aug 24 '18 at 18:44
  • $\begingroup$ Hint: Two Pythagorean triples you might know {5,12,13} and {9,12,15} $\endgroup$ – karakfa Aug 24 '18 at 18:58

Your mistake is in trying proportions in the first place!

In particular, you don't necessarily know that the two triangles are similar - there's no reason to assume the angles you've marked as the same are the same. In fact, the triangles you're hoping for as the answer: 5-12-13 and 9-12-15 aren't similar.

To get the correct answer, you can either use the Pythagorean theorem twice as you mention or use something like Heron's formula for the area first.

  • $\begingroup$ The altitude breaks the big triangle into two smaller right angles. I assumed from there that the remaining base angles had to be complementary to each other. Putting some values into the diagram helped me to see what I was doing wrong much better. Thank you. That also helped me to see why I was coming to an erroneous solution about the angles. (the altitude forming the angles was forced to be 90 degrees which is not the case). I was wrong to assume they were complementary just because both will have a complement to 90! $\endgroup$ – sec1 Aug 24 '18 at 19:05

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