Question

Prove that in any triangle the length of a side is less than half of the perimeter.

Answer 1

Lets consider a Triangle with area greater than Zero (obviously :/)
\[\sqrt{s(s-a)(s-b)(s-c)} = Area \implies Area^2 = s(s-a)(s-b)(s-c) \]
If Area > 0 \(\implies \) \(Area^2 > 0\).
\[s(s - a)(s-b)(s-c)> 0 \]
Note. s is the half perimeter.

Answer 2

I have never ever done this problem before ...So ... DO NOT TRUST ME.

Answer 3

Lets not square it, okay?
\[\sqrt{s(s-a)(s-b)(s-c)} > 0\]
s = (a+b+c)/2
\[\sqrt{2s(2s-2a)(2s-2b)(2s-2c)} > 0\]\[\sqrt{(a+b+c)(b + c - a)(a+b-c)(a+c - b)} >0\]
For a triangle, third side can not be greater than the sum of the other two sides.

Answer 4

is it right?

Answer 5

yes it is

Answer 6

what does area have to di with perimeter?

Answer 7

id prove it for an equitri and asses it from there

Answer 8

i see you used heron for relating sides ...

Answer 9

a+b>c
a+b+c>2c
(a+b+c)/2 > c

Answer 10

pssstt... i am still not sure if it's a conclusive proof, just made it out of thin air :/

Answer 11

oh foolformath completed it

Answer 12

how many assumptions are we allowed to make? fool used a thrm for a proposition i think

Answer 13

a + b > c
a+ b - c + c -c>0
2s - 2c > 0
s > c
same can be done for every side

Answer 14

Sum of any two sides of a triangle is greater than the third side.

http://www.proofwiki.org/wiki/Sum_of_Two_Sides_of_Triangle_Greater_than_Third_Side