﻿ Triangle Inequality Theorem: The rule explained with pictures and examples

# Triangle Inequality Theorem

Rule explained

#### Can any 3 side lengths form a triangle?

For instance, can I create a triangle from sides of length...say 4, 8 and 3 ?

# No! It's actually not possible!

As you can see in the picture below, it's not possible to create a triangle that has side lengths of 4 , 8 , and 3

It turns out that there are some rules about the side lengths of triangles. You can's just make up 3 random numbers and have a triangle! You could end up with 3 lines like those pictured above that cannot be connected to form a triangle.

### The Formula

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side.

Note: This rule must be satisfied for all 3 conditions of the sides.

In other words, as soon as you know that the sum of 2 sides is less than (or equal to ) the measure of a third side, then you know that the sides do not make up a triangle.

You can experiment for yourself using our free online triangle inequality theorem calculator -- which lets you enter any three sides and explains how the triangle inequality theorem applies to them.

#### Do I have to always check all 3 sets?

NOPE!

You only need to see if the two smaller sides are greater than the largest side!

Look at the example above, the problem was that 4 + 3 (sum of smaller sides) is not greater than 10 (larger side)

### Demonstrationsillustrating the Triangle Inequality Theorem

The interactive demonstration below shows that the sum of the lengths of any 2 sides of a triangle must exceed the length of the third side. The demonstration also illustrates what happens when the sum of 1 pair of sides equals the length of the third side--you end up with a straight line! You can't make a triangle!

Otherwise, you cannot create a triangle from the 3 sides.

##### Demonstration 1

When the sum of 1 pair of sides exactly equals the measure of a 3rd side.

##### Demonstration 2

When the sum of 1 pair of sides is less than the measure of a 3rd side.

### Practice Problems

No

Use the triangle inequality theorem and examine all 3 combinations of the sides. As soon as the sum of any 2 sides is less than the third side then the triangle's sides do not satisfy the theorem.

Yes

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 5 + 6 > 7
No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 1.2 + 1.6 $$\color{Red}{ \ngtr }$$ 3.1
No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 6 + 8 $$\color{Red}{ \ngtr }$$ 16
No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 5 + 5 $$\color{Red}{ \ngtr }$$ 10
Yes

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 7 + 9 >15

### Practice Problems II

difference < x < sum
7 -2 < x < 7+2

Answer: 5 < x < 9

difference < x < sum
12 -5 < x < 12 + 5

Answer: 7 < x < 17

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