Fill out the Problem-solving boxes to talk yourself through the problem. In the “DO” boxes below, fill in the
blank boxes, picking one number from the integers 1 through 10, at most one time each (cannot reuse any
numbers when picking a new side length) to complete the two scenarios below:
1. A triangle with the side lengths of these numbers that CAN NOT exist. (3pts)
7,11 ,?(1,2,3,4,5,6,7,8,9,10)
Understand & Think (1pt): What is being asked in the problem and what does that mean? What do I know and what does it
mean? What plan am I going to try?
Do/Answer (2pts): I will write out my response to the question,
explaining my answer and what it means. I will explain why my
answer makes sense. (Fill in the missing length in the empty box.)
Explanation of my answer and why it makes sense:
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.