Home
Determining the Range for the Third Side of a Triangle Using the Triangle Inequality Theorem

Determining the Range for the Third Side of a Triangle Using the Triangle Inequality Theorem

January 09, 2025 Education

The Triangle Inequality Theorem and Determining the Third Side of a Triangle

The triangle inequality theorem is a fundamental principle in geometry that helps determine the possible lengths of the third side of a triangle when the lengths of two sides are known. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let's explore how to use this theorem to find the range for the third side of a triangle with two given side lengths of 5 cm and 12 cm.

Applying the Triangle Inequality Theorem

Let's denote the side lengths of the triangle as follows:

a 5 cm b 12 cm c the length of the third side

According to the triangle inequality theorem, we need to satisfy the following conditions:

a b > c a c > b b c > a

Substituting the known values (a 5 cm, b 12 cm) into these inequalities, we get:

5 12 > c
c 5 c > 12
c > 7 cm 12 c > 5
c > -7 cm (This condition is always satisfied since c must be positive)

From the first two inequalities, we can deduce that the third side, c, must fall between 7 cm and 17 cm. Thus, the range for the length of the third side is:

boxed{7

Interpreting the Results

The outcome indicates that the third side of the triangle can have any number of values, but these values are bounded between 7 cm and 17 cm. Without additional information such as at least one angle, we cannot pinpoint an exact value, but we can definitively state the range within which the third side must lie.

It's important to note that the triangle inequality theorem not only helps in determining the possible lengths of the third side but also ensures the sides can form a triangle. The rule that the sum of any two sides must be greater than the third side is a direct application of this theorem.

Generalizing the Process

The same principle can be applied to other triangles with different side lengths. For example, if we have side lengths of 5 cm and 9 cm:

a 5 cm b 9 cm c the third side

Using the triangle inequality theorem, we find:

5 9 > c
c 5 c > 9
c > 4 cm 9 c > 5
c > -4 cm (This condition is always satisfied since c must be positive)

Thus, the range for the third side is:

boxed{4

Similarly, if we have a triangle with side lengths of 8 cm and 5 cm:

a 8 cm b 5 cm c the third side

Using the theorem, we find:

8 5 > c
c 8 c > 5
c > -3 cm (This condition is always satisfied since c must be positive) 5 c > 8
c > 3 cm

Hence, the range for the third side is:

boxed{3

Conclusion

The triangle inequality theorem is a powerful tool for determining the possible lengths of the third side of a triangle. By applying this principle, we can establish a range for the third side's length, ensuring that the sides can form a valid triangle. Understanding this concept is crucial for various applications in geometry and beyond.

Related Posts