Question

Two sides of a triangle have lengths 18 m and 23 m. Describe the possible lengths of the third side.

Answer 1

In order for the triangle inequality to hold, the third side must have a length between the sum and difference of the other two sides. Here, that means ...

... 23 m - 18 m < third side < 18 m + 23 m

... 5 m < third side < 41 m

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Comment on the triangle inequality

For sides a, b, c of a triangle, the triangle inequality may be written as either of

• a+b > c
• a+b ≥ c

This must apply to any mapping of side lengths to a, b, c. Some references write it one way; other references write it the other way. In the "equal to" case, the triangle is degenerate, having an area of zero.

Depending on which version of the triangle inequality you subscribe to, the "<" symbol in the above answer may be replaced by "≤". For the cases where the third side is 5 m or 41 m, the "triangle" will look like a line segment and will have an area of zero.