Geometry

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Triangle Inequality Theorem

What is the Triangle Inequality Theorem?

The triangle inequality theorem tells us that:
The sum of two sides of a triangle must be greater than the third side.
This theorem can be used to prove if a combination of three triangle side lengths is possible. See the image below for an illustration of the triangle inequality theorem.

triangle inequality theorem

Example Problem

Referencing sides x, y, and z in the image above, use the triangle inequality theorem to eliminate impossible triangle side length combinations from the following list.
1.) x = 2, y = 3, z = 5
2.) x = 5, y = 12, z = 13
3.) x = 3, y = 4, z = 5
4.) x = 12, y = 13, z = 27
5.) x = 2, y = 9, z = 12

Solution:
Side length combinations #1, #4, and #5 do not satisfy the requirements of the triangle inequality theorem and therefore are not possible.

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