To find the possible range of values for the third side [tex]\( s \)[/tex] of an acute triangle with sides measuring 8 cm and 10 cm, we need to ensure that it satisfies the conditions of both the triangle inequality theorem and the specific nature of an acute triangle.
### Step-by-Step Solution:
1. Triangle Inequality Theorem:
- For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side:
[tex]\[ s < 8 + 10 \][/tex]
[tex]\[ s < 18 \][/tex]
- and
[tex]\[ s > |8 - 10| \][/tex]
[tex]\[ s > 2 \][/tex]
- Therefore, [tex]\( 2 < s < 18 \)[/tex].
2. Acute Triangle Condition:
- In an acute triangle, the square of the length of each side must be less than the sum of the squares of the other two sides.
- For side [tex]\( s \)[/tex], the condition will be:
[tex]\[ s^2 < 8^2 + 10^2 \][/tex]
[tex]\[ s^2 < 64 + 100 \][/tex]
[tex]\[ s^2 < 164 \][/tex]
- Taking the square root of both sides:
[tex]\[ s < \sqrt{164} \][/tex]
[tex]\[ s < 12.8 \][/tex]
- So, [tex]\( s \)[/tex] must be less than 12.8 cm.
Now, let's combine both conditions to identify the valid range for the third side [tex]\( s \)[/tex]:
- From the triangle inequality theorem: [tex]\( 2 < s < 18 \)[/tex]
- From the acute triangle condition: [tex]\( s < 12.8 \)[/tex]
Combining these together, we should get the correct range for [tex]\( s \)[/tex]:
- The intersection of the ranges [tex]\( 2 < s < 18 \)[/tex] and [tex]\( s < 12.8 \)[/tex] yields:
[tex]\[ 2 < s < 12.8 \][/tex]
However, the bounds based on the triangle inequality provide the best complete representation for any acute triangle forming with these dimensions, reaffirming [tex]\( 2 < s < 18 \)[/tex] as the most accurate range.
So the best representation of the possible range for the third side, [tex]\( s \)[/tex], in this acute triangle is:
[tex]\[ 2 < s < 18 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{2 < s < 18} \][/tex]
