Determining the Range for the Third Side of a Triangle

Understanding the range of possible measures for the third side of a triangle, given the measures of the other two sides, is crucial in various fields including geometry, physics, and engineering. This article explores how to determine such a range using the triangle inequality theorem. We will apply this theorem to a given example where the other two sides of the triangle measure 7 and 11.

Triangle Inequality Theorem

The triangle inequality 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. This theorem helps us determine the possible range for the third side of a triangle when the lengths of the other two sides are known.

Applying the Theorem to Our Example

Given two sides of a triangle with lengths a 7 and b 11, we need to find the range of possible measures for the third side, denoted as c.

Using the triangle inequality theorem, the inequalities we need to consider are:

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

Substituting the given values:

7 c > 11 11 c > 7 7 11 > c

Solving the Inequalities

Let's solve each inequality step by step:

1. (7 c > 11)

Solving for (c):

[begin{align*} 7 c > 11 c > 11 - 7 c > 4 end{align*}]

2. (11 c > 7)

Solving for (c):

[begin{align*} 11 c > 7 c > 7 - 11 c > -4 end{align*}]

This condition is always satisfied since (c) must be positive.

3. (7 11 > c)

Solving for (c):

[begin{align*} 7 11 > c 18 > c c Combining the relevant inequalities, we find:

[begin{align*} 4 c 18 end{align*}]

Therefore, the range for the measurement of the third side (c) is 4 to 18.

Revisit and Explanation

Using the triangle inequality theorem:

(c 11 - 7' will always hold good say 4.1 (7 4.1 11.11 (Then 11 - 7 4) (7 11 11 4.1 15.1, 7 4.1 11.11 (11 7 18, 11 4.1 15.1, 7 4.1 11.11

From this, the range of possible measurements for the third side is [4, 18].

Summary

The key takeaways are:

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Validating the Range: If (x) is the measure of the third side, then: (x (x 7 > 11)Rightarrow x > 4) (x 11

Combining these, we find that the range of possible measurements for the third side is 4 to 18 units.

Conclusion

By applying the triangle inequality theorem, we have successfully determined the range of possible measures for the third side of a triangle given that the other two sides measure 7 and 11. This example demonstrates the practical application of this theorem in solving geometric problems and provides a clear understanding of how to determine the valid range of side measurements in a triangle.