Inequalities in Triangle

Triangle Inequality Theorem

Can we make a triangle?

Objectives

• Determine if 3 sides can form a triangle

• Find the range for a third side of a triangle given 2 sides

Triangle naming norms

• We use uppercase letters to indicate the angles (A, B, C).

• We use lowercase letters to indicate the sides (a, b, c).

• Side a is opposite angle A, side b is opposite angle B, side c is opposite angle C.

What if the sum EQUALS the other side?

Example - if the side lengths were 1, 2 and 3, they would NOT form a triangle. The sum of two smaller sides must be GREATER than the longest side.

What it looks like when the sum of the two smaller sides is NOT larger than the third side.

What it looks like when the sum of the two smaller sides is NOT larger than the third side.

Multiple Choice

Can these sides form a triangle? 2, 9, 5

• Yes

• No

Multiple Choice

Can these sides form a triangle? 6, 4, 4

• Yes

• No

Multiple Choice

Can these sides form a triangle? 22, 22, 22

• Yes

• No

Multiple Choice

Can these sides form a triangle?

$$6\ \frac{1}{4},\ 14\ \frac{1}{2},\ 6\ \frac{1}{2}$$  

• Yes

• No

If we know two sides of a triangle, we can find a range of possible lengths for the third side.

• ADD the two given sides to get the MAXIMUM length of the third side.

• SUBTRACT the two given sides to get the MINIMUM length of the third side.

The two given sides of a triangle are 6 and 9. What could the third side be?

Consider that the 6 and 9 could be the 2 larger sides OR the 6 and 9 could be the 2 smaller sides.

Open Ended

Name a possible length for the third side of a triangle with sides 4 and 7.

Open Ended

Name a possible length for the third side of a triangle with sides 11 and 19.

Multiple Select

Which of the following could be the third side of a triangle with sides 5.2 and 7.2? Check all that apply.

• 21

• 2

• 9

• 12.4

• 11

Multiple Select

Which of the following could NOT be the third side of a triangle with sides 35 and 14? Check all that apply.

• 21

• 48

• 26

• 35

• 14

Multiple Choice

If the measures of 2 sides of a triangle are 57 and 32, what is the LEAST possible measure of the third side if the measure is an integer?

• 25

• 26

• 88

• 89