*Sometimes I get a bug in my head about something and it won't go away until I write it down and work it out. This time I recorded it in a journal, so I decided to upload it here on a non-comic days. Comments are welcome, especially from my students.*

A few years back, I got tired of every right triangle problem involving a 3-4-5, 6-8-10, 30-40-50 or 5-12-13 triangle. I started creating a list of other triples that I could use when avoiding irrational numbers.

*First discovery:*Triples are either E

^{2}+ O

^{2}= O

^{2}or E

^{2}+ E

^{2}= E

^{2}

(where E is an even number, and O is an odd number), and the latter form could always be reduced to the first form. For need of a term, let's say that the triples that are reduced are in the simplest form, which means that the numbers are relatively prime.

*Second discovery:*Many (a, b, c) triples could be reduced to (a, b, b + 1), when a was odd, or (a, b, b + 2), when a was even. (There are some that don't fit either, but those will have to wait.)

Squares can be made by summing consecutive odd numbers. When that addend is a perfect square, the result is a

**Pythagorean Triple**.

9 = 3

^{2}==> 3

^{2}+ 4

^{2}= 5

^{2}

25 = 5

^{2}==> 5

^{2}+ 12

^{2}= 13

^{2}

Was there a quicker way to find the triples for 7, 9, 11, etc?

I'll get to that.

Also notice that 31 + 33 = 64, which is 8

^{2}, so 8

^{2}+ 15

^{2}= 17

^{2}.

And it had been under my nose all the time that 4

^{2}+ 3

^{2}= 5

^{2}.

The simplest Pythagorean Triple of them all, fits both these models.

*(to be continued)*

### Questions for my students

*Extra credit for answering or for participating in the discussion.*

- Can you explain in your own words what I meant by "relatively prime"? What do you know about
*prime*numbers that might give you a hint? - Can you find Pythagorean Triples where the smallest side is 9, 11, and 13?

Remember: 9-12-15 doesn't count. It can be reduced to 3-4-5. - Why can't there be any Triples of the form: Even
^{2}+ Even^{2}= Odd^{2}?

## 1 comment:

1.Relatively prime numbers are the ones which don't have any factor in common , i.e their HCF is 1. :)

2. 9^2 + b^2 = (b+1)^2 ==> 2b+1 = 81

==> b = 40 , so triple is 9,40,41

3. E^2 = E, and E + E is always Even

MY QUESTION: so 9,40,41 will be the only triple with 9 as the shortest side ?

I'm preparing for MBA entrance exams like GMAT and CAT, so studying from that perspective (solve big problems in minimum time).

If I wanted to find all the triples with 9 as the shortest side, how to do it?

:) I really loved your post on Pythagorean triples (this post) and i found out the triple so easily. I'm glad.

Thanks a lot :)

Will be eagerly waiting for your response.

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