To determine the possible range of values for the third side [tex]\( s \)[/tex] of an acute triangle with the two given sides measuring [tex]\( 8 \, \text{cm} \)[/tex] and [tex]\( 10 \, \text{cm} \)[/tex], we need to take into account several key geometric constraints and properties.
### Steps to determine the constraints:
1. Triangle Inequality Theorem
For any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
a + b > c
\][/tex]
Therefore, in our case with sides [tex]\( 8 \)[/tex], [tex]\( 10 \)[/tex], and [tex]\( s \)[/tex]:
[tex]\[
8 + 10 > s \implies s < 18
\][/tex]
Additionally,
[tex]\[
8 + s > 10 \implies s > 2
\][/tex]
and
[tex]\[
10 + s > 8 \implies s > -2
\][/tex]
The most restrictive of these is [tex]\( s > 2 \)[/tex].
2. Acute Triangle Condition
For the triangle to be acute, the square of the longest side must be less than the sum of the squares of the other two sides. If [tex]\( s \)[/tex] is the longest side:
[tex]\[
s^2 < 8^2 + 10^2
\][/tex]
Calculating the right hand side:
[tex]\[
8^2 + 10^2 = 64 + 100 = 164
\][/tex]
Therefore:
[tex]\[
s^2 < 164
\][/tex]
Taking the square root:
[tex]\[
s < \sqrt{164} \approx 12.8
\][/tex]
3. Combine these inequalities:
From the triangle inequality, we have [tex]\( s < 18 \)[/tex]. From the acute triangle condition, we have [tex]\( s < 12.8 \)[/tex]. Therefore, the condition [tex]\( s < 12.8 \)[/tex] is more restrictive.
4. Ensuring all constraints:
Checking the lower bound:
- From the triangle inequality, we have: [tex]\( s > 2 \)[/tex].
Checking the upper bound:
- From the acute triangle condition: [tex]\( s < 12.8 \)[/tex].
Putting all the constraints together, we get:
[tex]\[
2 < s < 12.8
\][/tex]
### Conclusion
The best representation of the possible range of values for the third side [tex]\( s \)[/tex] in an acute triangle with the given side lengths is:
[tex]\[
6 < s < 12.8
\][/tex]
