To determine the possible range of values for the third side, [tex]\( s \)[/tex], of an acute triangle where the two given sides are 8 cm and 10 cm, we need to use the properties of an acute triangle.
In an acute triangle, all angles are less than 90 degrees. For a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] (where [tex]\( c \)[/tex] is the largest side), the following conditions must hold true:
1. [tex]\( a^2 + b^2 > c^2 \)[/tex]
2. [tex]\( a^2 + c^2 > b^2 \)[/tex]
3. [tex]\( b^2 + c^2 > a^2 \)[/tex]
Given:
- [tex]\( a = 8 \)[/tex]
- [tex]\( b = 10 \)[/tex]
Let's determine the range for the third side, [tex]\( s \)[/tex], where [tex]\( s \)[/tex] will be the largest side. We will examine two main points:
### 1. When [tex]\( s \)[/tex] is the largest side:
Using the first condition for the largest side:
[tex]\[ 8^2 + 10^2 > s^2 \][/tex]
[tex]\[ 64 + 100 > s^2 \][/tex]
[tex]\[ 164 > s^2 \][/tex]
[tex]\[ s < \sqrt{164} \][/tex]
[tex]\[ s < 12.806248474865697 \][/tex]
### 2. When [tex]\( s \)[/tex] isn't the largest side:
Now, we need to consider the other conditions where 8 or 10 could be the largest sides, but since we are only asked for the feasible range for [tex]\( s \)[/tex], we'll focus on:
[tex]\[ s > |a - b| \][/tex]
[tex]\[ s < a + b \][/tex]
From:
[tex]\[ s \geq |8 - 10| \][/tex]
[tex]\[ s > 2 \][/tex]
Finally,
[tex]\[ s < a + b \][/tex]
[tex]\[ s < 18 \][/tex]
### Conclusion:
Considering both conditions, to ensure the angles remain acute, the possible range for [tex]\( s \)[/tex] is:
[tex]\[ 6 < s < 12.8 \][/tex]
Thus, the best representation of the possible range of values for the third side, [tex]\( s \)[/tex], is:
[tex]\[ 6 < s < 12.8 \][/tex]
So the correct answer is:
[tex]\[ \boxed{6
