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Oculus8596

Banned

From: Great Lakes,Illinois

Registered: 2024-09-18

Posts: 126

Right Triangle Prove

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.

Last edited by Oculus8596 (2024-09-27 02:44:15)

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Bob

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Registered: 2010-06-20

Posts: 10,627

Re: Right Triangle Prove

hi Oculus8596

Welcome to the forum.

opposite integer :- the number that is the same distance from zero but with the opposite sign.

This restriction is unnecessary as the result follows even when m and n are any real numbers.

Start with  a^2 + b^2 = (m^2 - n^2)^2 + 4m^2n^2

Continue with the algebra until you reach c^2

Pythagoras theorem works 'in reverse' so a^2 + b^2 = c^2 is sufficent to prove the right angle.

Bob

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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

Oculus8596

Banned

From: Great Lakes,Illinois

Registered: 2024-09-18

Posts: 126

Re: Right Triangle Prove

hi Oculus8596

Welcome to the forum.

opposite integer :- the number that is the same distance from zero but with the opposite sign.

This restriction is unnecessary as the result follows even when m and n are any real numbers.

Start with  a^2 + b^2 = (m^2 - n^2)^2 + 4m^2n^2

Continue with the algebra until you reach c^2

Pythagoras theorem works 'in reverse' so a^2 + b^2 = c^2 is sufficent to prove the right angle.

Bob

The algebra is a bit tedious but doable.

---

The best things in life are not always free.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Right Triangle Prove

It's just two more lines to the result.

Bob

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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

Oculus8596

Banned

From: Great Lakes,Illinois

Registered: 2024-09-18

Posts: 126

Re: Right Triangle Prove

It's just two more lines to the result.

Bob

(m^2 - n^2)^2 + 4m^2n^2 = c^2

(m^2 - n^2)(m^2 - n^2) + 4m^2n^2 = c^2

m^4 - 2m^2n^2 + n^4 + 4m^2n^2 = c^2

m^4 + 2m^2n^2 + n^4 = c^2

Where do we go from here?

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The best things in life are not always free.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Right Triangle Prove

That's it. In any triangle if a^2 + b^2 = c^2 then you may conclude that the triangle is right angled.

Bob

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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

Oculus8596

Banned

From: Great Lakes,Illinois

Registered: 2024-09-18

Posts: 126

Re: Right Triangle Prove

That's it. In any triangle if a^2 + b^2 = c^2 then you may conclude that the triangle is right angled.

Bob

Wonderful. Work is completed. I thought the right side needed to look like the left side of a^2 + b^2 = c^2.

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The best things in life are not always free.