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Pythagorean Triples

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Joined: 17 Apr 2012
Posts: 4

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22 Apr 2012, 22:26
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.
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Joined: 02 Sep 2009
Posts: 51185

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22 Apr 2012, 22:33
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GanAiwen wrote:
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.

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.
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24 Apr 2012, 06:31
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GanAiwen wrote:
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 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:
http://en.wikipedia.org/wiki/Pythagorea ... an_triples
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11 Aug 2017, 01:04
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Re: Pythagorean Triples &nbs [#permalink] 11 Aug 2017, 01:04
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Pythagorean Triples

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