# Triangle Inequality Theorem

How to use the triangle inequality theorem to determine if the given numbers can be the sides of a triangle: formula, 4 examples, and their solutions.

## Formula

### Formula

The sides of a triangle a, b, c

satisfy this inequality:

a + b > c.

a + b: Sum of the shorter sides

c: Longest side

This is the triangle inequality theorem.

## Example 1

### Example

### Solution

The longest side is 4.

The shorter sides are 2 and 3.

Then see if

2 + 3 > 4

is true.

If this inequality is true,

then these three numbers

can be the sides of a triangle.

2 + 3 = 5

So 5 > 4.

This is true.

2 + 3 > 4

is true.

So 2, 3, 4 can form a triangle.

(= can be the sides of a triangle.)

So [Can form a triangle] is the answer.

### Figure

This is the triangle

formed by the sides 2, 3, and 4.

2 + 3 > 4

is true.

So 2, 3, 4 can form a triangle.

## Example 2

### Example

### Solution

The longest side is 9.

The shorter sides are 2 and 7.

Then see if

2 + 7 > 9

is true.

2 + 7 = 9

So 9 > 9.

This is false.

2 + 7 > 9

is false.

So 2, 7, 9 cannot form a triangle.

(= can't be the sides of a triangle.)

So [Cannot form a triangle] is the answer.

### Figure

This figure shows that

why 2, 7, 9 cannot form a triangle.

2 + 7 > 9

is false.

(2 + 7 = 9)

So 2, 7, 9 cannot form a triangle.

## Example 3

### Example

### Solution

The longest side is 8.

The shorter sides are 3 and 4.

Then see if

3 + 4 > 8

is true.

3 + 4 = 7

So 7 > 8.

This is false.

3 + 4 > 8

is false.

So 3, 4, 8 cannot form a triangle.

So [Cannot form a triangle] is the answer.

### Figure

This figure shows that

why 3, 4, 8 cannot form a triangle.

3 + 4 > 8

is false.

(3 + 4 < 8)

So 3, 4, 8 cannot form a triangle.

## Example 4

### Example

### Solution

The longest side is 5.

(Choose one of the 5s.)

The shorter sides are 3 and 5.

Then see if

3 + 5 > 5

is true.

3 + 5 = 8

So 8 > 5.

This is true.

3 + 5 > 5

is true.

So 3, 5, 5 can form a triangle.

So [Can form a triangle] is the answer.

### Figure

This is the triangle

formed by the sides 3, 5, and 5.

3 + 5 > 5

is true.

So 3, 5, 5 can form a triangle.