Final answer:
The third side of a triangle with sides seven and nine can be any length greater than 2 but less than 16. Therefore, options A (8), B (5), and D (13) are possible lengths for the third side, while option C (22) is not possible according to the Triangle Inequality Theorem.
Step-by-step explanation:
To find the possible lengths for the third side of a triangle with two sides of lengths seven and nine, we must apply 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 remaining side. Therefore, we can set up two inequalities to find the range for the length of the third side (let's call it x):
- 7 + 9 > x
- 7 + x > 9
- 9 + x > 7
From the first inequality, we learn that x must be less than 16. From the second and third inequalities, we find that x must be greater than 2 and 1 respectively, which are both true for any x greater than 2. Combining these two we get 2 < x < 16.
Therefore, the length of the third side must be greater than 2 but less than 16. Checking the provided options:
- A. 8 - Possible, since it is between 2 and 16
- B. 5 - Possible, since it is between 2 and 16
- C. 22 - Not possible, since it is greater than 16
- D. 13 - Possible, since it is between 2 and 16