Triangle Inequality Theorem

Overview

Not every set of three lengths can form a triangle. The two shorter pieces have to be able to reach each other when they are hinged to the ends of the longest piece — if their sum is less than the longest piece they fall short, and if their sum is exactly equal they lie flat on top of it instead of popping up into a triangle. The Triangle Inequality Theorem turns that picture into a test, and the same test produces the possible range for a third side when the other two sides are known.

Vocabulary

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side; for three given lengths, if the sum of the two smaller lengths is greater than the longest length, the measurements will form a triangle.

Third Side Range

Given two sides of a triangle, the third side must measure more than the difference of the two sides and less than their sum.

Segment

A part of a line that consists of two endpoints and all the points between them; segments can only meet at their endpoints, which is why three lengths must be long enough to reach one another to form a triangle.

Third Side

The remaining side of a triangle when the other two sides are given; its possible lengths are exactly what the Triangle Inequality Theorem restricts.

Main Ideas

1. To test three lengths, add the two smaller ones: the sum must be strictly greater than the longest length.
2. Equality fails — if the two smaller lengths add to exactly the longest length, the pieces lie flat and no triangle forms.
3. Given two sides a and b, the third side x must satisfy (larger − smaller) < x < (larger + smaller).
4. Segments can only meet at their endpoints, which is why the two shorter pieces must be long enough to reach each other and pop up into a triangle.

Lesson Notes

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Lesson Video

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Practice Problems

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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.

• Triangle Inequality Theorem — Practice Worksheet (PDF)Solutions (PDF)