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In what other ways can you use the Triangle Inequality Theorem?

What if you are given the dimensions of a triangular shape but not shown a picture? How would use the Triangle Inequality Theorem to show that the inequalities hold true? If you are given all three sides as numbers instead of labels on a diagram, you simply need to substitute the numbers into the inequalities and verify that they are still true.

For example, suppose you're give this information: 

\(\small\mathsf{AB}\) = 40, \(\small\mathsf{BC}\) = 45, and \(\small\mathsf{AC}\) = 29.4

How can you use the Triangle Inequality Theorem to verify that this set of numbers is a triangle?
Generally, you ask this kind of question for each arrangement of numbers: Is \(\small\mathsf{AB}\) + \(\small\mathsf{BC}\) > \(\small\mathsf{AC}\)? Your first verification would look like this: 

Given \(\small\mathsf{AB}\) = 40, \(\small\mathsf{BC}\) = 45 and \(\small\mathsf{AC}\) = 29.4
Find out if \(\small\mathsf{AC}\) < \(\small\mathsf{AB}\) + \(\small\mathsf{BC}\)
29.4 ?< 40 + 45
29.4 is < 85 so the first inequality holds true.

Question

What about the second inequality - is \(\small\mathsf{BC}\) < \(\small\mathsf{AB}\) + \(\small\mathsf{AC}\) given \(\small\mathsf{AB}\) = 40, \(\small\mathsf{BC}\) = 45 and \(\small\mathsf{AC}\) = 29.4? Use the same steps as in the example above to verify this inequality. Then click the button to check your work.

Find out if \(\small\mathsf{BC}\) < \(\small\mathsf{AB}\) + \(\small\mathsf{AC}\)
45? < 40 + 29.4
45 < 69.4, so the second inequality holds true.

Question

Is the last inequality true, \(\small\mathsf{AB}\) < \(\small\mathsf{BC}\) + \(\small\mathsf{CA}\) given \(\small\mathsf{AB}\) = 40, \(\small\mathsf{BC}\) = 45 and \(\small\mathsf{AC}\) = 29.4?

Find, \(\small\mathsf{AB}\) < \(\small\mathsf{BC}\) + \(\small\mathsf{AC}\)
So, 40 <? 45 + 29.4
40 < 74.4 so the third inequality holds true as well.

Since you were able to verify the inequalities for all three arrangements of segments, or sides, you can say with certainty that ABC is a triangle. Remember, in order for the Triangle Inequality Theorem to hold true, ALL of the combinations of sides have to work. So, that means all three inequalities have to be true. If just one of them fails, then the numbers you were given do not represent a triangle.