Building blocks

Triangle Side Inequalities

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

1) AB + BC > AC
2) AB + AC > BC
3) AC + BC > AB

Notice in the figure that as vertex C moves closer and closer to side AB, the sum of the lengths of sides AC and CB gets smaller.

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Can it ever reach the point where AC+CB=AB? Yes, but then it is no longer a triangle. It would be a straight line!. If AC + CB is not greater than AB, then the sides are not longer enough to make a triangle.

Is a triangle with these three sides possible?
a. 12, 13, 21
b. 7, 8, 15

Solution:
a. 25>21. Yes.
b. 15 = 15. No.

A triangle has two sides of length 4cm and 12 cm. Describe the possible length of the third side, x.

Solution:
There are several possibilities:

But all of them must verify the triangle side inequalities:

x+4>12, then x>8
x+12>4, then x>-8
4+12>x, then 16>x

Consequently 16 > x > 8