Question on uniqueness of 3-4-5, 5-12-13 and 8-15-17

[anand.quantum]

Hi
The MGMAT geometry strategy guide wants us to remember the pythagorean triples.

Had a question about these, let us take say the 3-4-5 triangle. If I were to have a right triangle, whose "shortest" side is 3, is the only right triangle I can form a 3-4-5? If the answer is yes, what is a sure shot way to verify, if the answer is no, what is the best way to verify.

In other words is 3-4-5 the "ONLY UNIQUE" combination of perfect squares that meets the pythagorean requirement of

a^2 + b^2 = c^2

Similarly you can extend the question for the other sides of 3-4-5 and the other two combinations 5-12-13 and 8-15-17.

Reason I am asking this is, if I know one side of any combination of the pythagorean triplets, if they are unique, I can deduce the other two.

[RonPurewal]

If you are specifically talking about side lengths that are whole numbers, then, yes, 3/4/5 is the only right triangle that has 3 as a side.
as far as verifying this -- you don't need to know any fancy number theory; you can just try all the feasible combinations. i.e., if 3^2 + b^2 = c^2, then that means 9 + b^2 = c^2.
so, you are looking for two perfect squares that have a difference of 9 between them.
it's pretty quick to see that 16 and 25 are the only nonzero perfect squares that have this difference. (as soon as you get past 25, the difference between any two squares is always too big.)

--

On the other hand, if you're not talking about whole numbers, then there are infinitely many possibilities:
3/1/√10
3/√2/√11
3/√3/√12
3/2/√13
etc.

--

by the way, "only unique" is redundant.

[RonPurewal]

by the way, there's nothing GMAT-specific about this question, so you can find much more comprehensive information by just Googling "pythagorean triples" and reading a bunch of the pages you see.
and you won't have to wait for a response, either. yay internet!

(there's no way we could really do this topic justice in a single forum post, if you are looking for the kind of depth that you seem to be asking about here.)

[anand.quantum]

Hi Ron

Thanks. Actually reason I asked is because of the following post ( from 2008 or so) on MGMAT forum mgmat-cat-1-similar-triangles-t3642.html

I see that one of the sides of the right triangle was 12. I wanted to know if I can "iterate" through all possible combinations to get to the desired length ratio - I was not able to :). I ended up plugging answers and got to the solution. I dont know how the author of the problem above got the respective sides. It would be nice if you can walk me through the solution.

[RonPurewal]

Hi Ron

Thanks. Actually reason I asked is because of the following post ( from 2008 or so) on MGMAT forum mgmat-cat-1-similar-triangles-t3642.html

I see that one of the sides of the right triangle was 12. I wanted to know if I can "iterate" through all possible combinations to get to the desired length ratio - I was not able to :). I ended up plugging answers and got to the solution. I dont know how the author of the problem above got the respective sides. It would be nice if you can walk me through the solution.

hi,
please post your question on the thread you linked to. thanks.

we don't want multiple threads about the same problem; that will just make everything harder for everyone.
especially when you've already located an existing thread!