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.