A triangle has two sides of length 18 and 4. What is the smallest possible whole-number length for the third side?

Explanation:

Let a = 4 and b = 18 be the two known sides of the triangle.

Let c be the remaining side

We'll use a version of the triangle inequality theorem to set up the possible range of values for c.

The range of values for c is determined by the following

b-a < c < b+a

18-4 < c < 18+4

14 < c < 22

Telling us that the third side c is between 14 and 22, excluding both endpoints. This means c = 14 and c = 22 are not allowed. If either value were to happen, then the triangle becomes a single line instead. Try it out with slips of paper.

Because c = 14 is not allowed, we must go to the next whole number up. Therefore, c = 15 is the smallest possible length for the third side, when the third side is a whole number.

If the third side wasn't a whole number, then we could have something like c = 14.1 or c = 14.01 or c = 14.001 or c = 14.0001, and so on. We could steadily get closer and closer to 14, but never actually arrive to it. In short, there is no "smallest third side". This is likely why your teacher is focusing on the whole numbers so there is a smallest value of c possible.

Extra info: As you can probably guess, the largest side possible is c = 21, which is one less than 22. Like above, this is only possible if c is a whole number.

step-by-step explanation:

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