Final Answer:
The difference between the largest and smallest possible lengths of the third side of the triangle is d) 25 units.
Step-by-step explanation:
In a triangle, the third side's length must be greater than the difference between the other two sides and less than the sum of the other two sides. Applying the triangle inequality theorem, the smallest possible length of the third side is |16 - 9| = 7 units (taking the difference between the given sides). The largest possible length of the third side is 16 + 9 = 25 units (taking the sum of the given sides). Thus, the difference between these values gives us 25 - 7 = 18 units as the range of possible lengths for the third side. However, the question asks for the difference specifically, which is 25 - 7 = 18 units.
In this scenario, the triangle inequality theorem holds true. For any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Given two sides measuring 9 and 16 units, the third side must adhere to this rule. The smallest length of the third side occurs when it is the difference between the given sides, resulting in 7 units.
Conversely, the largest length is the sum of the given sides, totaling 25 units. Subtracting the smallest from the largest yields a difference of 18 units, showing the possible range for the third side's length. Therefore, the correct answer to the question regarding the difference in lengths of the third side is indeed 18 units.