To determine the possible range for the third side, [tex]\( s \)[/tex], in a triangle given that the other two sides measure 8 cm and 10 cm, we must use the triangle inequality theorem. The triangle inequality theorem states that for any triangle, the sum of any two sides must be greater than the third side.
Let's denote the sides of the triangle as [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], where:
- [tex]\( a = 8 \)[/tex] cm
- [tex]\( b = 10 \)[/tex] cm
- [tex]\( c \)[/tex] is the third side.
We need to determine the range for [tex]\( c \)[/tex] (which we are calling [tex]\( s \)[/tex]).
1. According to the triangle inequality, the sum of any two sides must be greater than the third side. This gives us three inequalities:
- [tex]\( a + b > c \)[/tex]
- [tex]\( a + c > b \)[/tex]
- [tex]\( b + c > a \)[/tex]
2. Plugging in the given values:
- [tex]\( 8 + 10 > c \)[/tex] simplifies to [tex]\( 18 > c \)[/tex] or [tex]\( c < 18 \)[/tex].
- [tex]\( 8 + c > 10 \)[/tex] simplifies to [tex]\( c > 2 \)[/tex].
- [tex]\( 10 + c > 8 \)[/tex] simplifies to [tex]\( c > -2 \)[/tex]. However, this is always true for positive [tex]\( c \)[/tex] and thus does not give a new constraint.
3. Therefore, combining the relevant constraints:
- [tex]\( 2 < c < 18 \)[/tex]
Hence, the best representation of the possible range of values for the third side [tex]\( s \)[/tex] in this triangle is:
[tex]\[
2 < s < 18
\][/tex]
So, the correct answer is:
[tex]\[\$2
