To determine the possible range of values for the third side [tex]\( s \)[/tex] of a triangle, we use the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given the two sides of the triangle measuring 8 cm and 10 cm, let's denote these sides as [tex]\( a = 8 \)[/tex] and [tex]\( b = 10 \)[/tex]. The third side is [tex]\( s \)[/tex].
The triangle inequality conditions are:
1. [tex]\( a + b > s \)[/tex]
2. [tex]\( a + s > b \)[/tex]
3. [tex]\( b + s > a \)[/tex]
Substituting [tex]\( a = 8 \)[/tex] and [tex]\( b = 10 \)[/tex]:
1. [tex]\( 8 + 10 > s \)[/tex]
[tex]\[ 18 > s \][/tex]
[tex]\[ s < 18 \][/tex]
2. [tex]\( 8 + s > 10 \)[/tex]
[tex]\[ s > 2 \][/tex]
3. [tex]\( 10 + s > 8 \)[/tex]
This inequality simplifies to:
[tex]\[ s > -2 \][/tex]
Since [tex]\( s > -2 \)[/tex] is always true for positive values of [tex]\( s \)[/tex], we do not need to consider it further.
Combining the first two conditions, we get the range:
[tex]\[ 2 < s < 18 \][/tex]
Thus, the best representation of the possible range of values for the third side [tex]\( s \)[/tex] is:
[tex]\[ 2 < s < 18 \][/tex]
Therefore, the correct answer is:
[tex]\[ 2 < s < 18 \][/tex]
