The Wonderful World of the Triangle Inequality Theorem

Step-by-step Guide: The Triangle Inequality Theorem

1. The Theorem Stated: For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. If we have a triangle with sides \(a\), \(b\), and \(c\), then the following must be true:
   \( a + b > c \\
   b + c > a \\
   a + c > b \)
2. How to Use the Theorem: To determine if three lengths can form a triangle, simply check if they satisfy the above inequalities.

Examples

Example 1:
Can sides of lengths \(5 \text{ cm}\), \(7 \text{ cm}\), and \(10 \text{ cm}\) form a triangle?

Solution:
Using the theorem, let’s check:

For sides \(5 \text{ cm}\) and \(7 \text{ cm}\):
\( 5 \text{ cm} + 7 \text{ cm} = 12 \text{ cm} \)
Since \(12 \text{ cm} > 10 \text{ cm}\), the first condition is met.

For sides \(5 \text{ cm}\) and \(10 \text{ cm}\):
\( 5 \text{ cm} + 10 \text{ cm} = 15 \text{ cm} \)
Since \(15 \text{ cm} > 7 \text{ cm}\), the second condition is met.

For sides \(7 \text{ cm}\) and \(10 \text{ cm}\):
\( 7 \text{ cm} + 10 \text{ cm} = 17 \text{ cm} \)
Since \(17 \text{ cm} > 5 \text{ cm}\), the third condition is met.

All conditions are met, so yes, these sides can form a triangle!

Example 2:
Can sides of lengths \(4 \text{ cm}\), \(8 \text{ cm}\), and \(12 \text{ cm}\) form a triangle?

Solution:
Using the theorem:

For sides \(4 \text{ cm}\) and \(8 \text{ cm}\):
\( 4 \text{ cm} + 8 \text{ cm} = 12 \text{ cm} \)
Since \(12 \text{ cm} = 12 \text{ cm}\), this does not meet the “greater than” condition.

Thus, the sides cannot form a triangle!

Practice Questions:

1. Can sides of lengths \(4 \text{ cm}\), \(7 \text{ cm}\), and \(13 \text{ cm}\) form a triangle?
2. Can sides of lengths \(9 \text{ cm}\), \(11 \text{ cm}\), and \(20 \text{ cm}\) form a triangle?
3. Can sides of lengths \(15 \text{ cm}\), \(17 \text{ cm}\), and \(31 \text{ cm}\) form a triangle?

Answers:

1. No
2. No
3. Yes