Suppose you were given a measurement of 14 inches for one side of a triangle and 10 inches for another side, but no information about the third side. Based on the Triangle Inequality Theorem, what could you say about the missing side?
We know from the theorem that the third side (call it x because it is unknown) is shorter than the sum of the two known sides. We can express this inequality this way: x < 14 + 10, or we can say that x must be less than 24. We can also say that 14 < x + 10, which means that 4 < x or x > 4.
Notice how we do not say that 10 < x + 14 because this would yield -4 < x or x > -4. If x were greater than -4, the numbers from 0 to 4 would actually not work to form a triangle. So, as a rule, when solving for a missing side, use the inequalities that involve 1) the two other sides and 2) the missing side with the shorter of the two other sides.
Once we know that x should be between 4 and 24, can we check the answer? Absolutely. All we have to do is pick a number less than 24 and seeing if it would work in our theorem.
| Pick a number between 4 and 24, since 24 > x > 4. | Let's choose 20. |
| Visually inspect the combination 20, 14, 10. | Test to see if the set of numbers satisfies the Triangle Inequality Theorem (any side must be shorter than the sum of the other two sides). |
| Ask if there is a counterexample for the theorem. | Can you find a number in the set 20, 14, and 10 that breaks this rule? |
| Draw a conclusion. | Since there is no counterexample, we can feel sure that 20, 14, and 10 would form a triangle--thereby verifying that the missing side x must be less than 24. |
