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Triangle Inequality Theorem

Learning about theorems is an excellent way to load up our brains with all kinds of easy tips and tricks for solving even the most complex math problems. A theorem is something that we already know is true, and we can apply them quickly and easily. One such theorem is the triangle inequality theorem. But what does this rule tell us about triangles? How do we use it? Let's find out:

The triangle inequality theorem defined

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

  • If we add two sides of a triangle together, the result is always greater than the length of the third side

In other words:

a + b > c

a + c > b

b + c > a

Under the triangle inequality theorem, all of these statements are true.

Visualizing the triangle inequality theorem

It helps if we can visualize the triangle inequality theorem. Take a look at this triangle:

Without doing any math at all, we can see that the triangle inequality theorem is true for this drawing of this triangle. Take all 3 sides and imagine laying them flat, the lengths of any two lines side by side appear longer than the remaining line.

Working with the triangle inequality theorem

We can use the triangle inequality theorem to solve a number of math problems.

Let's say we have a theoretical triangle with sides that measure 7, 9, and 13 units.

But wait.. Is such a triangle even possible?

Let's find out using the triangle inequality theorem:

7 + 9 = 16

9 + 13 = 22

7 + 13 = 20

All of these results are larger than the longest side of the triangle, which is 13. Therefore, we can safely say that this is in fact a possible triangle.

Now let's assume that we have a theoretical triangle with sides that are 4, 8, and 15 units long. Is this triangle possible, or are we dealing with an imposter?

Let's find out:

4 + 8 = 12

This value is less than the greatest side of the triangle. Therefore, we know right away that this triangle just isn't possible.

What would a failed triangle look like?

It helps to visualize what might happen if a theoretical figure does not satisfy the requirements of the triangle inequality theorem.

If one side is longer than the sum of two other sides, we would see a gap instead of a vertex. The triangle wouldn't be complete -- just like a pyramid without its capstone.

But what happens if one side is completely equal to the sum of the two other sides? This theoretical shape would still fail the requirements of the triangle inequality theorem. Why? Because we would be left with a completely straight line.

Try it yourself by drawing a triangle with one side 10 cm long and two other sides with lengths of 5cm. There's no way to accomplish this impossible task, as the only possible result is a straight line. We call this triangle a "degenerate triangle" because its area is zero.

The triangle inequality theorem works for all triangles

Unlike many other theorems that work only for right triangles or oblique triangles, the triangle inequality theorem works for all triangles.

When we work with right triangles, the triangle inequality theorem is valid due to the principles of another theorem -- the Pythagorean theorem.

When we work with oblique or "general" triangles, the triangle inequality theorem is valid due to the law of cosines.

The converse of the triangle inequality theorem

Note that the converse of the triangle inequality theorem is also true:

If we have three numbers and each is less than the sum of the two others, then we can use these numbers to create a real triangle with a positive area.

Topics related to the Triangle Inequality Theorem

Triangles

30-60-90 Triangles

Volume of a Pyramid

Flashcards covering the Triangle Inequality Theorem

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Triangle Inequality Theorem

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

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