To find the range of the possible values for the third side, [tex]\(s\)[/tex], of an acute triangle given two sides measuring 8 cm and 10 cm, we need to use the triangle inequality theorem.
### 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. This forms three inequalities:
1. [tex]\(a + b > s\)[/tex]
2. [tex]\(a + s > b\)[/tex]
3. [tex]\(b + s > a\)[/tex]
Here, given [tex]\(a = 8\)[/tex] cm and [tex]\(b = 10\)[/tex] cm, we need to find the valid range for [tex]\(s\)[/tex].
### Step-by-Step Solution:
1. First Inequality:
- [tex]\(a + b > s\)[/tex]
- [tex]\(8 + 10 > s\)[/tex]
- [tex]\(18 > s\)[/tex]
- Therefore, [tex]\(s < 18\)[/tex]
2. Second Inequality:
- [tex]\(a + s > b\)[/tex]
- [tex]\(8 + s > 10\)[/tex]
- [tex]\(s > 2\)[/tex]
3. Third Inequality:
- [tex]\(b + s > a\)[/tex]
- [tex]\(10 + s > 8\)[/tex]
- [tex]\(s > -2\)[/tex]
- This inequality is always true since [tex]\(s > 2\)[/tex] is more restrictive.
### Synthesis of Inequalities:
Combining the valid ranges obtained from the inequalities, we get:
[tex]\[2 < s < 18\][/tex]
### Conclusion:
The best representation of the possible values for the third side [tex]\(s\)[/tex] of the triangle is:
[tex]\[ \boxed{2 < s < 18} \][/tex]
