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22 Apr 2012, 23:26
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So I'm trying to recall the following: can you deduce that a triangle is a Pythagorean Triple (PT) based on only the hypotenuse? Ex. If we know it's a right triangle, and are given the hypotenuse 5, can we assume that the legs are 3 and 4?
I know this doesn't work the other way around, meaning if we knew it was a right triangle, and was only given a leg of 3, we couldn't assume it was a PT. Is it a different case for a hypotenuse though?
Would appreciate any help, as well as a link to where you might have found your explanation.
I know this doesn't work the other way around, meaning if we knew it was a right triangle, and was only given a leg of 3, we couldn't assume it was a PT. Is it a different case for a hypotenuse though?
Would appreciate any help, as well as a link to where you might have found your explanation.
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22 Apr 2012, 23:33
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GanAiwen
Knowing that hypotenuse equals to 5 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 3:4:5. Or in other words: if \(a^2+b^2=5^2\) DOES NOT mean that \(a=3\) and \(b=4\), certainly this is one of the possibilities but definitely not the only one. In fact \(a^2+b^2=5^2\) has infinitely many solutions for \(a\) and \(b\) and only one of them is \(a=3\) and \(b=4\).
For example: \(a=1\) and \(b=\sqrt{24}\) or \(a=2\) and \(b=\sqrt{21}\)...
Check Triangles chapter of Math Book for more: math-triangles-87197.html
Hope it's clear.
24 Apr 2012, 07:31
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GanAiwen
I will add one more point here:
Pythagorean triplets represent the right triangle with integral sides. So if you know the hypotenuse is 5 and the other 2 sides are integers too, then the other two sides must be 3 and 4. Else, they could be anything.
Check out this wikipedia link for more:
https://en.wikipedia.org/wiki/Pythagorea ... an_triples
