Question

A triangle has side lengths 9 and 15. Which of the following could be the length of the third side? Select all that apply.
(2 points)
Responses

5
5

7
7

18
18

23
23

27

Answer 1

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 \( c \):

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

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 \( c \):

1. \( 9 + 15 > c \)
- This simplifies to \( c < 24 \).

2. \( 9 + c > 15 \)
- This simplifies to \( c > 6 \).

3. \( 15 + c > 9 \)
- 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**.