Pythagorian triple

[Roadrunner]

Is there a way to tell tht 8, 15, and 17 is a Pythagorian triple without calculating?

 

In this particular problem DS knew one side was 8*3 =24, another side was 15*3=45, so my kid solved for the third side the long way, but didn't know 8,15,17 is a triple. Are we suppose to memorize some of those triples, or is there a way to tell.

 

He does understand 3:4:5

[elmerRex]

Hope that I am saying this right: No.

 

Here is why: There are infinite numbers of "Pythagorean Triplets" it will be impossible for most students or teachers to tell--without effort if it is a PT they are looking at, but there are a couple of patterns to them as a whole.

 

Tripletss are always one of two types--or almost always one of these two types, I do not know an exception, but I am not a math master.

Type 1--ALL 3 are even

Type 2--2/3 are odd and 1/3 is even.

 

It can be much fun to talk about WHY these patterns exist.

[regentrude]

I don't know whether there is an easy way to tell that 3 numbers for a Pythagorean triple other than trying it out.

 

You can construct triples by using Euclid's formula: if you take two integers, m>n, then a=m^2-n^2, b= 2mn, and c=m^2+n^2 form a triple.

For 8, 15, 17, m=4 and n=1.

So, if he student knows this, he could try to see if he can identify a pair of m,n for his three numbers. Note that this only works for primitive triples; if the numbers share a common factor, you first need to separate that factor (like in all triples that consist of three even numbers).

 

I'd make sure students recognize the 3-4-5 triple because that occurs a lot. Beyond that, nope.

[wendyroo]

When I was an SAT tutor, we taught students to look for the triples 3, 4, 5 and 5, 12, 13 because those were the ones that showed up most frequently on the test and could save a student considerable time.

 

Wendy

[Roadrunner]

O.K. That's good to know. He can spot 3-4-5 easily, but didn't spot 8,15,17. This was aops preA problem and the solution used the triple to solve it. It sounded to me that DS should have somehow been able to recognize it, so I am glad that isn't so. I might just add some of those triples to our "math poster." :)

[kiana]

The only ones that stick in my memory are 3:4:5, 5:12:13, 7:24:25, 8:15:17

 

BUT -- if he knows that one side is 3a and another side is 3b, he should be able to reason through and get that that the third side must be 3c, where a:b:c are the sidelengths of a right triangle.

[Roadrunner]

.

[Luckymama]

Dd has seen 7:24:25 so many times this year in AoPS precalc and in prepping for the SAT Math 2 subject test.

[Roadrunner]

Dd has seen 7:24:25 so many times this year in AoPS precalc and in prepping for the SAT Math 2 subject test.

 

I am going to write some of those out and put them on the wall.