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# Extremely Simple question

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Do you know why the difference of two sides of a triangle is always smaller than the third side? I have been trying to prove that a - b < c for a few hours :( Tnx in advance

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2 sides of triangle ABC is always longer or equal(in degenerate cases) than third side.That's because third side is a line and line is a shortest path between 2 points,so any other path is longer.
Let's sides lengths is A,B,C.

A+B>=C
subtract B from both sides
A>=C-B
Proven.

i'd guess, too simple to be homework.

[Edited by - Dmytry on September 2, 2004 11:51:33 AM]

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Quote:
 Original post by Dmytry2 sides of triangle ABC is always longer or equal(in degenerate cases) than third side.That's because third side is a line and line is a shortest path between 2 points,so any other path is longer.Let's sides lengths is A,B,C.A+B>=Csubtract B from both sidesA>=C-BProven.i'd guess, too simple to be homework.

Indeed, too simple to be homework

tnx

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Are you using different letters here?
He said 'c > b-a', you said 'a>c-b'.

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does not matter. As about why, because i started writing with a+b>c ,he started writing with a-b<c , note the order abc.
edit: and, OP, don't feel too bad that you haven't solved. I remembered that in the school there i studied, most of the class was unable to prove other very simple teorem... so proving this may be as well be too hard for homework :).

[Edited by - Dmytry on September 3, 2004 1:06:52 AM]

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Quote:
 Original post by Dmytrydoes not matter. As about why, because i started writing with a+b>c ,he started writing with a-b

You're just a gifted person, Dmytry ;)

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Quote:
 Original post by Dmytry2 sides of triangle ABC is always longer or equal(in degenerate cases) than third side.That's because third side is a line and line is a shortest path between 2 points,so any other path is longer.Let's sides lengths is A,B,C.A+B>=Csubtract B from both sidesA>=C-BProven.i'd guess, too simple to be homework.

Hmm, how does that PROVE it? Your first line is what youre trying to prove, and you use it in your proof?

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use the Pythagorean theorem. I think its postulate that each side length of a triangle is > 0.

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Quote:
 Do you know why the difference of two sides of a triangle is always smaller than the third side? I have been trying to prove that a - b < c for a few hours :(

Quote:
Original post by healeyx76
Quote:
 Original post by Dmytry2 sides of triangle ABC is always longer or equal(in degenerate cases) than third side.That's because third side is a line and line is a shortest path between 2 points,so any other path is longer.Let's sides lengths is A,B,C.A+B>=Csubtract B from both sidesA>=C-BProven.i'd guess, too simple to be homework.

Hmm, how does that PROVE it? Your first line is what youre trying to prove, and you use it in your proof?

First line:
"2 sides of triangle ABC is always longer or equal(in degenerate cases) than third side"
- it's comes from definition of straight line, that line is a shortest path.As i said "That's because third side is a line and line is a shortest path between 2 points,so any other path is longer"
Then i wrote equation of what i said
A+B>=C

And OP asked to prove
that difference of two sides is always smaller than third side.
I subtracted b from both sides of equation, and got a difference,that luckly was smaller than other side.

Formal proof would be even shorter:
"
by definition of line:
a+b>=c
subtract b from both sides
a>=c-b
"

edit:and yes,it maybe would be bit better if i said something like

"a ,by definition of line,is a shortest path between it's endpoints, and b+c is also the path between endpoints, so it's longer or equal to a.
a<=b+c
subtracting b from both sides
a-b<=c
"
what is, indeed, exactly The Same Thingtm, and i just improved my grammar/order of abc.

[Edited by - Dmytry on September 3, 2004 1:28:37 PM]

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