Triangle Inequality Theorem

What Is the Triangle Inequality Theorem?

The Triangle Inequality Theorem is a fundamental rule in geometry that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This rule ensures that the three sides can connect to form a closed shape with three corners, or vertices. If this condition isn't met, the sides won't be able to reach each other to form a triangle.

Imagine you have three sticks. If you have two very short sticks and one very long one, you can't connect their ends to make a triangle. The two short sticks, laid end-to-end, would just lie flat along the long stick. The theorem gives us a mathematical way to test this before we even start drawing.

For any triangle with side lengths labeled as $a$, $b$, and $c$, the theorem can be expressed as three distinct inequalities:

• $a+b>c$
• $a+c>b$
• $b+c>a$

For a set of three lengths to form a triangle, all three of these statements must be true.

Why Does This Theorem Make Sense? A Visual Intuition

The core idea behind the theorem is something you already know intuitively: the shortest distance between two points is a straight line.

Let's think about a triangle as three points: Point A, Point B, and Point C. The side lengths are the distances between these points.

Imagine you need to walk from Point A to Point C. The direct path is the line segment connecting them, which has a length we can call $b$. Now, what if you decided to walk from Point A to Point B first (a distance of $c$), and then from Point B to Point C (a distance of $a$)? You've essentially taken a detour. Your total walking distance, $a+c$, will definitely be longer than the direct path, $b$. Therefore, $a+c>b$.

This logic applies to any pair of points. The distance between B and C (side $a$) must be shorter than going from B to A and then A to C (a path of $c+b$). This simple, real-world concept is the foundation of the Triangle Inequality Theorem. The only way the paths would be equal is if Point B were directly on the line segment between A and C, which would result in a straight line, not a triangle!

How to Use the Triangle Inequality Theorem: The Test

To determine if three side lengths can form a triangle, you need to perform a simple test. Let's say you are given three side lengths, $a$, $b$, and $c$.

The Formal Test: Check all three inequalities.

If every single one of these is true, then you can form a triangle. If even one of them is false, it's impossible.

The Shortcut Test: There's a faster way! You only need to perform one check.

Why does this shortcut work? Because it's the toughest test. If the two shortest sides can team up to be longer than the biggest side, the other two combinations will automatically be true. For example, adding a short side to the long side will obviously be greater than the other short side. This shortcut saves you time and is the most common way to apply the theorem.

How Do You Find the Range of a Possible Third Side?

A very common and useful application of the theorem is to find the possible range of lengths for a third side when you already know two sides. Let's say you have two sides of a triangle, $a$ and $b$, and you want to find all the possible lengths for the third side, $c$.

We can use the three inequalities from the theorem to figure this out:

1. From $a+b>c$, we know the third side $c$ must be less than the sum of the other two sides.
2. From $a+c>b$, we can rearrange it by subtracting $a$ from both sides to get $c>b-a$.
3. From $b+c>a$, we can rearrange it to get $c>a-b$.

Combining the second and third points tells us that the third side $c$ must be greater than the difference between the other two sides. (We use the positive difference, which is written as $|a-b|$).

This gives us a powerful two-part rule to find the range of the third side, $c$.

This compound inequality tells you the minimum and maximum boundaries for the third side.

Common Mistakes to Avoid

The Triangle Inequality Theorem is straightforward, but a few common slip-ups can lead to the wrong answer. Here are the main ones to watch out for.

• Forgetting the "Greater Than" Rule: The theorem requires the sum of two sides to be strictly greater than (>) the third. An answer like $10=10$ is not good enough. As we saw in Example 3, if the sum of the two shorter sides equals the longest side, they form a flat line, not a triangle.
• Only Checking One Combination: If you don't use the shortcut, you must check all three inequalities. For sides $4,8,3$, checking $4+8>3$ (True) isn't enough. You must also check $4+3>8$ (False). Using the shortcut (is $3+4>8$?) prevents this mistake.
• Mixing Up Sum and Difference for the Range Problem: When finding the possible length of a third side, students sometimes forget whether to add or subtract. Remember, the third side is squeezed between the difference and the sum.

Here is a table showing the correct vs. incorrect thinking for finding the range of a third side, given sides of $10$ and $4$:

Quick Reference Summary

Here are the key takeaways for the Triangle Inequality Theorem in a nutshell.

• The Main Rule: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
• The Best Shortcut: To quickly check if three lengths form a triangle, just confirm that the sum of the two shorter sides is greater than the longest side.
• The "Equals" Case: If the sum of the two shorter sides equals the longest side (e.g., $3,4,7$), the lengths form a flat line, not a triangle. This is called a degenerate triangle.
• Finding a Missing Side's Range: If you know two sides ($a$ and $b$), the third side ($c$) must be between their difference and their sum: $|a-b|<c<a+b$.