Prove that lengths a,b,c > 0 with a^2 +b^2 =c^2 actually fit together to make a triangle
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Prove that lengths a,b,c > 0 with a^2 +b^2 =c^2 actually fit together to make a triangle

[From: ] [author: ] [Date: 11-10-11] [Hit: ]
actual triangle, but those should be easier.......
(hint: show that a+b>c)

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This one is easiest to approach using proof by contradiction: suppose a + b <= c and show this forces one of the hypotheses to fail.

Now what to do with that? How about squaring both sides. You get

a^2 + 2ab + b^2 <= c^2. Using the second hypothesis and subtracting equal quantities from both sides gives 2ab <= 0, but that is impossible if a, b > 0. Therefore our supposition is false and we have a + b > c.

Now that doesn't finish the problem, you still need to see that a + c > b and b + c > a for there to be an
actual triangle, but those should be easier.
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