To determine the range of possible values for the third side of an acute triangle with the given sides measuring 10 cm and 16 cm, we need to use the triangle inequality theorem. The triangle inequality theorem states that for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
Let's denote the unknown side by [tex]\(x\)[/tex]. We will now apply the triangle inequality theorem:
1. [tex]\(10 + 16 > x\)[/tex]
[tex]\[ 26 > x \][/tex]
[tex]\[ x < 26 \][/tex]
2. [tex]\(10 + x > 16\)[/tex]
[tex]\[ x > 16 - 10 \][/tex]
[tex]\[ x > 6 \][/tex]
3. [tex]\(16 + x > 10\)[/tex]
[tex]\[ x > 10 - 16 \][/tex]
This inequality is also covered by [tex]\(x > 6\)[/tex].
Taking all these inequalities together, the range of possible values for [tex]\(x\)[/tex] is:
[tex]\[ 6 < x < 26 \][/tex]
Therefore, the best description of the range of possible values for the third side [tex]\(x\)[/tex] of the triangle is:
[tex]\[ \boxed{6 < x < 26} \][/tex]
