To determine the possible side lengths of the third side of a triangle with known side lengths of 6 and 12, we can 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 remaining side.
Let's denote the lengths of the sides of the triangle as [tex]\( a = 6 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c \)[/tex] is the unknown third side. The triangle inequality theorem gives us three conditions that must all be satisfied:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
We can use these conditions to find the range of possible values for [tex]\( c \)[/tex].
### Step-by-Step Solution:
1. First, let's calculate the inequality for [tex]\( a + b > c \)[/tex]:
[tex]\[
6 + 12 > c \implies 18 > c \implies c < 18
\][/tex]
2. Next, let's calculate the inequality for [tex]\( a + c > b \)[/tex]:
[tex]\[
6 + c > 12 \implies c > 6
\][/tex]
3. Finally, let's calculate the inequality for [tex]\( b + c > a \)[/tex]:
[tex]\[
12 + c > 6 \implies c > -6
\][/tex]
However, this last inequality [tex]\( c > -6 \)[/tex] is not stricter than the previous condition that [tex]\( c > 6 \)[/tex]. Therefore, the relevant condition here is simply [tex]\( c > 6 \)[/tex].
Combining the results from the above inequalities:
[tex]\[
6 < c < 18
\][/tex]
Thus, the possible side lengths for the third side [tex]\( c \)[/tex] satisfy the condition [tex]\( 6 < c < 18 \)[/tex].
### Conclusion:
The correct answer is:
D) [tex]\( 6 < c < 18 \)[/tex]
