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.
7,11 ,?(1,2,3,4,5,6,7,8,9,10)
Understand & Think : 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: 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:
The problem is asking for a triangle with side lengths of 7, 11, and an unknown side length that cannot exist.
To determine if a triangle with these side lengths can exist, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's apply this theorem to the given side lengths:
7 + 11 > ? (unknown side length)
18 > ? (unknown side length)
Since we do not know the value of the unknown side length, we cannot determine if it fulfills the triangle inequality theorem. However, we can see that there are many possible values for the unknown side length (1, 2, 3, 4, 5, 6, 7, 8, 9, or 10) that can make this triangle valid. Therefore, there is no integer value for the unknown side length that cannot exist.
Therefore, any integer value between 1 and 10 (inclusive) can be the missing length in the triangle with side lengths 7, 11, and ?.