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Pre-Algebra

Students discover and prove the triangle inequality theorem, and apply the theorem to determine if it is possible to form a triangle when given the lengths of three line segments.

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After completing this tutorial, you will be able to complete the following:

- State the triangle inequality theorem.
- Prove the triangle inequality theorem
- Apply the triangle inequality theorem.

The triangle inequality theorem for Euclidean geometry essentially states that the shortest distance between two points is a straight line. This intuitive concept is one of Euclid's axioms in his text, Elements. Formally, the theorem can be stated as follows.

Triangle Inequality Theorem: In a triangle, one side length is less than the sum of the other two side lengths.

If a, b, and c are the lengths of the sides of a triangle, then the theorem gives the following inequalities:

While there are multiple proofs of the triangle inequality theorem, the proof Euclid used relies on is the side-angle relationship for triangles. This relationship states that if one angle of a triangle has greater measure than a second angle, then the side opposite the first is longer than the side opposite the second. The proof follows.

So, the triangle inequality theorem provides an upper bound for the length of any side of a triangle. We can use this theorem to derive another theorem that will give us a lower bound.

Notice that because in Figure 1, it is also true that we also have that This means that and we have a lower bound for c. In a similar manner, we can find lower bounds for a and b. Formally, we have the following theorem.

Reverse Triangle Inequality Theorem: In a triangle, one side length is greater than the absolute value of the difference of the other two side lengths.

If a, b, and c are the lengths of the sides of a triangle, then the theorem gives the following inequalities:

Thus, the triangle inequality and the reverse triangle inequality give us a range for the length of a side of a triangle:

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Students should know the definition of a triangle; know the side angle relationship in a triangle; know the definition and properties of inequality; be able to find the length of a line segment; be able to draw line segments with a given length; and use the notation for a line segment between two points and its length. |

Course | Pre-Algebra |

Type of Tutorial | Visual Proof |

Key Vocabulary | angle, inequality, relationship |