The Triangle Inequality Theorem

In Δ TAB (Figure ), if T, A, and B represent three points on a map and you want to go from T to B, going from T to A to B would obviously be longer than going directly from T to B. The following theorem expresses this idea.

Figure 1 Two paths from T to B.

Theorem 38 (Triangle Inequality Theorem): The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Example 1: In Figure 2, the measures of two sides of a triangle are 7 and 12. Find the range of possibilities for the third side.

Figure 2 What values of x will make a triangle possible?

Using the Triangle Inequality Theorem, you can write the following:

7 + x > 12, so x > 5

7 + 12 > x, so 19 > x (or x < 19)

Therefore, the third side must be more than 5 and less than 19.