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Geometry Tool Belt: The Triangle Inequality

Geometry Tool Belt: The Triangle Inequality

Author: c o

Introduce the triangle inequality and prove that it holds for all triangles.

Provide examples of applying the triangle inequality to determine which lengths can and cannot be used to form triangles.

The triangle inequality is one of the most useful facts to remember, not just in geometry, but in all of mathematics. It crops up in numerous fields in a variety of forms and is therefore a useful fact to remember. After mentioning some necessary background knowledge, the triangle inequality is then defined and proved. Following the proof are some examples of applying the triangle inequality.

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Before you dive into this lesson, you should know a little about triangles.

Of course you know what a triangle is, but do you know about isosceles triangles?

                                                                   isoscelese triangle

The most important facts to remember about an isosceles triangle are that two sides have equal length and therefore two angles have equal measure.

Another important fact to remember, and one that is true of all triangles, is that the meausre of an angle is relative to the length of its opposing side.  Here is a picture to make this clearer:


In the above picture, angle a is bigger than angle b, and angle b is bigger than angle c.  Likewise, side A is longer than side B, and side B is longer than side C.  In general, the longest side is opposite the largest angle, the sortest side is opposite the shortest angle.

The Triangle Inequality

The triangle inequality states that the sum of any two sides of a triangle is greater than the length of the remaining side.

If we have a triangle with sides A B and C, and we pick two of the sides, say C and B, then C+B > A.  

One nice corollary of the triangle inequality has to do with choosing lengths from which triangles can be constructed.  If for any three numbers we can find some two of them that when summed are less than or equal to the third, then we know that we cannot construct a triangle with those numbers as the side lengths.

For example 4, 2, and 1.  There is no triangle with one side length 4, another length 2, and the other length 1.  Why not? because 1 + 2 < 4.

And that's about all there is to understanding the basics of the triangle inequality.  Now for the proof!

Euclid's Proof of the Triangle Inequality

This video shows the proof of the triangle inequality as given by Euclid in The Elements, Book I Proposition XX.

The video is a little blurry at the start, but it soon comes into focus.

Source: Euclid's Elements, Colin O'Keefe

A little quiz

Which of the following triplets do not describe the side lengths of some triangle?


A) 1,1,1

B) 10, 3, 5

C) 30, 40, 50

D) 9, 8, 17


Answers: B and D

B doesn't work because 10 > 3 + 5

D doesn't work because 8 + 9 = 17, and the triangle inequality states the the sum of the two sides must be greater than the sum of the third.