To determine the possible lengths of the third side of a triangle with known side lengths of 5 and 8, we can use the triangle inequality theorem. This 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. Additionally, the length of the third side must be greater than the absolute difference of the other two sides.
Let's denote the sides of the triangle as [tex]\( a = 5 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c \)[/tex] as the unknown third side. According to the triangle inequality theorem:
1. The sum of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] must be greater than [tex]\( c \)[/tex]:
[tex]\[
a + b > c \implies 5 + 8 > c \implies 13 > c \implies c < 13
\][/tex]
2. The sum of [tex]\( a \)[/tex] and [tex]\( c \)[/tex] must be greater than [tex]\( b \)[/tex]:
[tex]\[
a + c > b \implies 5 + c > 8 \implies c > 3
\][/tex]
3. The sum of [tex]\( b \)[/tex] and [tex]\( c \)[/tex] must be greater than [tex]\( a \)[/tex]:
[tex]\[
b + c > a \implies 8 + c > 5 \implies c > -3 \quad (\text{This condition is automatically satisfied if } c > 3)
\][/tex]
Combining these inequalities, we get:
[tex]\[
3 < c < 13
\][/tex]
Thus, the possible range for the length of the third side [tex]\( c \)[/tex] is between 3 and 13 (exclusive).
Therefore, the correct answer is:
D) [tex]\( 3 < c < 13 \)[/tex]
