To determine the range of possible values for the third side of a triangle, it's important to apply the triangle inequality theorem. 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 denote the unknown third side as [tex]\( x \)[/tex].
Given the two sides of the triangle are 10 cm and 16 cm:
1. The first inequality: The sum of the lengths of the given sides must be greater than the length of the third side.
[tex]\[
10 + 16 > x
\][/tex]
Simplifying this, we get:
[tex]\[
26 > x \quad \text{or} \quad x < 26
\][/tex]
2. The second inequality: The sum of the unknown side and the smallest given side must be greater than the length of the largest given side.
[tex]\[
x + 10 > 16
\][/tex]
Simplifying this, we get:
[tex]\[
x > 6
\][/tex]
3. The third inequality: The sum of the unknown side and the largest given side must be greater than the length of the smallest given side.
[tex]\[
x + 16 > 10
\][/tex]
This inequality simplifies trivially:
[tex]\[
x > -6
\][/tex]
Since [tex]\( x \)[/tex] is a length and must be positive, we don't need to consider this result specifically.
Combining the valid inequalities, we get:
[tex]\[
6 < x < 26
\][/tex]
Thus, the range of possible values for the third side of the triangle is described by the inequality:
[tex]\[
6 < x < 26
\][/tex]
So, the best description of the range of the possible values for the third side of the triangle is:
[tex]\[
6 < x < 26
\][/tex]
