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## #1 2023-09-18 17:13:04

harpazo1965
Member
Registered: 2022-09-19
Posts: 183

### Sides of A Right Triangle

College Algebra
Section R.3

Suppose that m and n are positive integers with m > n.
If a = (m^2 - n^2), b = 2mn, and c = (m^2 + n^2), show that a, b, and c are the lengths of the sides of a right triangle.

Note: This formula can be used to find the sides of a right triangle that are integers such as 3, 4, 5; 5, 12, 13; and so on. Such triplets of integers are called Pythagorean triplets.

NOTE: Looking for the set up only.

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## #2 2023-09-19 04:30:10

Bob
Registered: 2010-06-20
Posts: 9,731

### Re: Sides of A Right Triangle

You need to show that two of these when squared and added make the same result as the third when squared.

But which to choose? I can see that c is bigger than a.  So I suggest working out c^2 and write that down. Then get an expression for a^2 + b^2. If this comes to the same as c^2 then you are done.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #3 2023-09-19 08:37:48

e_jane_aran
Novice
Registered: 2023-09-15
Posts: 2

### Re: Sides of A Right Triangle

harpazo1965 wrote:

Suppose that m and n are positive integers with m > n.
If a = (m^2 - n^2), b = 2mn, and c = (m^2 + n^2), show that a, b, and c are the lengths of the sides of a right triangle.

Note: This formula can be used to find the sides of a right triangle that are integers such as 3, 4, 5; 5, 12, 13; and so on. Such triplets of integers are called Pythagorean triplets.

NOTE: Looking for the set up only.

I suspect that they're expecting you to notice how the given side-length expressions hint of connections to perfect-square trinomiais.

If you add two of the above expressions, you get the third of the above expressions. Which pair works?

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## #4 2023-09-19 08:39:57

e_jane_aran
Novice
Registered: 2023-09-15
Posts: 2

### Re: Sides of A Right Triangle

I apologize for my failure at formatting in the above. I though [ imath ] tags would work. Kindly please ignore the "\qquad" part at the beginning of each line. D'oh!

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## #5 2023-09-19 10:36:38

harpazo1965
Member
Registered: 2022-09-19
Posts: 183

### Re: Sides of A Right Triangle

Bob wrote:

You need to show that two of these when squared and added make the same result as the third when squared.

But which to choose? I can see that c is bigger than a.  So I suggest working out c^2 and write that down. Then get an expression for a^2 + b^2. If this comes to the same as c^2 then you are done.

Bob

Thank you, Bob.

Last edited by harpazo1965 (2023-09-23 01:00:12)

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## #6 2023-09-19 19:08:02

Bob
Registered: 2010-06-20
Posts: 9,731

### Re: Sides of A Right Triangle

to both of you:

I have edited the above post so that it displays properly.  The correct command is square brackets math

When a member makes a post, it is visible to anyone who logs in to the site.  And anyone can respond to any post.  If someone disobeys our rules then I will take action. This may involve any of the following: Issuing a warning; Editing the offending part of a post; Deleting the post; Banning for a limited period; Removing the person completely from the membership. This last means that all that person's posts get deleted too.

At the moment I can see no reason to do any of these things.  Let's keep it that way.

Best wishes,

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #7 2023-09-23 01:00:57

harpazo1965
Member
Registered: 2022-09-19
Posts: 183

### Re: Sides of A Right Triangle

Ok. No problem. Let's get back to math.

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## #8 2023-09-23 01:11:44

harpazo1965
Member
Registered: 2022-09-19
Posts: 183

### Re: Sides of A Right Triangle

e_jane_aran wrote:
harpazo1965 wrote:

Suppose that m and n are positive integers with m > n.
If a = (m^2 - n^2), b = 2mn, and c = (m^2 + n^2), show that a, b, and c are the lengths of the sides of a right triangle.

Note: This formula can be used to find the sides of a right triangle that are integers such as 3, 4, 5; 5, 12, 13; and so on. Such triplets of integers are called Pythagorean triplets.

NOTE: Looking for the set up only.

I suspect that they're expecting you to notice how the given side-length expressions hint of connections to perfect-square trinomiais.

If you add two of the above expressions, you get the third of the above expressions. Which pair works?

If I add a^2 + c^2, the middle term cancels out. This does not leave me with much to play with.

If I do that, I am left with m^4 + n^4 twice. In other words, I get
2(m^4 + n^4) = 2m^4 + 2n^4, which does not represent b^2.

Last edited by harpazo1965 (2023-09-23 01:12:18)

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## #9 2023-09-23 16:37:56

amnkb
Member
Registered: 2023-09-19
Posts: 43

### Re: Sides of A Right Triangle

harpazo1965 wrote:

Suppose that m and n are positive integers with m > n.
If a = (m^2 - n^2), b = 2mn, and c = (m^2 + n^2), show that a, b, and c are the lengths of the sides of a right triangle.

Note: This formula can be used to find the sides of a right triangle that are integers such as 3, 4, 5; 5, 12, 13; and so on. Such triplets of integers are called Pythagorean triplets.

NOTE: Looking for the set up only.

e_jane_aran wrote:

I suspect that they're expecting you to notice how the given side-length expressions hint of connections to perfect-square trinomiais.

If you add two of the above expressions, you get the third of the above expressions. Which pair works?

harpazo1965 wrote:

If I add a^2 [and] c^2, the middle term cancels out. This does not leave me with much to play with.

If I do that, I am left with m^4 + n^4 twice. In other words, I get
2(m^4 + n^4) = 2m^4 + 2n^4, which does not represent b^2.

so maybe try a different pair of terms? (there r 2 other pairs; 1 of them works)

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## #10 2023-09-23 17:25:44

harpazo1965
Member
Registered: 2022-09-19
Posts: 183

### Re: Sides of A Right Triangle

amnkb wrote:
harpazo1965 wrote:

Suppose that m and n are positive integers with m > n.
If a = (m^2 - n^2), b = 2mn, and c = (m^2 + n^2), show that a, b, and c are the lengths of the sides of a right triangle.

Note: This formula can be used to find the sides of a right triangle that are integers such as 3, 4, 5; 5, 12, 13; and so on. Such triplets of integers are called Pythagorean triplets.

NOTE: Looking for the set up only.

e_jane_aran wrote:

I suspect that they're expecting you to notice how the given side-length expressions hint of connections to perfect-square trinomiais.

If you add two of the above expressions, you get the third of the above expressions. Which pair works?

harpazo1965 wrote:

If I add a^2 [and] c^2, the middle term cancels out. This does not leave me with much to play with.

If I do that, I am left with m^4 + n^4 twice. In other words, I get
2(m^4 + n^4) = 2m^4 + 2n^4, which does not represent b^2.

so maybe try a different pair of terms? (there r 2 other pairs; 1 of them works)

If I add a^2 + b^2 I get c^2. So, the addition of the two legs of this triangle
yields the value of the hypotenuse. You say?

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