To determine the range of possible values for the third side of the triangle, we need to use the 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.
Given the two sides of the triangle [tex]\(a = 10 \, \text{cm}\)[/tex] and [tex]\(b = 16 \, \text{cm}\)[/tex], we'll denote the unknown third side as [tex]\(x\)[/tex].
The triangle inequality theorem provides us with three inequalities:
1. [tex]\(a + b > x\)[/tex]
2. [tex]\(a + x > b\)[/tex]
3. [tex]\(b + x > a\)[/tex]
Substituting the given values:
1. [tex]\(10 + 16 > x\)[/tex]
2. [tex]\(10 + x > 16\)[/tex]
3. [tex]\(16 + x > 10\)[/tex]
Let's simplify these inequalities one by one:
1. [tex]\(10 + 16 > x\)[/tex] simplifies to [tex]\(26 > x\)[/tex], or [tex]\(x < 26\)[/tex].
2. [tex]\(10 + x > 16\)[/tex] simplifies to [tex]\(x > 6\)[/tex].
3. [tex]\(16 + x > 10\)[/tex] simplifies to [tex]\(x > -6\)[/tex], which is always true since side lengths are positive.
Combining the inequalities [tex]\(x < 26\)[/tex] and [tex]\(x > 6\)[/tex], we get:
[tex]\[6 < x < 26\][/tex]
Therefore, the range of possible values for the length of the third side [tex]\(x\)[/tex] in this triangle is:
[tex]\[\boxed{6 < x < 26}\][/tex]
Hence, the correct description is:
[tex]\[6 < x < 26\][/tex]
