Understand & Think: The problem is asking for a triangle with side lengths that cannot exist. To have a valid triangle, the sum of any two sides must be greater than the third side. I know that a triangle with side lengths 7 and 11 can exist, so I need to find a number that cannot be combined with either 7 or 11 to form a valid triangle.
Do/Answer:
To find the missing length, I will start by finding the possible ranges for the third side length. For a triangle with side lengths a, b, and c, the possible range for c is: |a - b| < c < a + b.
For a triangle with sides of lengths 7 and 11, the possible range for the third side is:
7 - 11 < c < 7 + 11
-4 < c < 18.
I need to find a number that falls outside this range, meaning it cannot be used as the third side length for a triangle with sides of lengths 7 and 11.
The only integer that falls outside this range is -5. Therefore, a triangle with side lengths 7, 11, and -5 cannot exist.
Explanation of my answer and why it makes sense:
A triangle with side lengths of 7, 11, and -5 cannot exist because -5 is less than 7 - 11, which violates the triangle inequality theorem. The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. In this case, 7 + (-5) is less than 11, which is not possible in a valid triangle.