To determine the possible lengths of the third side of a triangle with the given sides of lengths 9 and 15, we use the triangle inequality theorem. This theorem states that for any triangle with side lengths \( a \), \( b \), and \( c \):
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let’s denote the lengths of the sides of the triangle as \( a = 9 \), \( b = 15 \), and \( c \) as the unknown third side. We can apply the triangle inequality to find the range of possible values for \( c \):
-
\( 9 + 15 > c \)
- This simplifies to \( c < 24 \).
-
\( 9 + c > 15 \)
- This simplifies to \( c > 6 \).
-
\( 15 + c > 9 \)
- This condition is always satisfied as \( c > -6 \), which does not impose any new constraints.
Combining these two inequalities, we find:
- \( 6 < c < 24 \)
Now, we can check each of the given options to see which one(s) fall within this range:
- 5: Not valid, since \( 5 < 6 \).
- 7: Valid, since \( 6 < 7 < 24 \).
- 18: Valid, since \( 6 < 18 < 24 \).
- 23: Valid, since \( 6 < 23 < 24 \).
- 27: Not valid, since \( 27 > 24 \).
Thus, the lengths that could be the length of the third side are 7, 18, and 23.