To determine whether a triangle can be formed with the given side lengths of 4 inches, 12 inches, and 17 inches, we need to check the triangle inequality theorem. The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
We will verify three conditions to see if they hold true:
1. The sum of the lengths of side1 and side2 must be greater than the length of side3:
[tex]\(4 + 12 > 17\)[/tex]
2. The sum of the lengths of side1 and side3 must be greater than the length of side2:
[tex]\(4 + 17 > 12\)[/tex]
3. The sum of the lengths of side2 and side3 must be greater than the length of side1:
[tex]\(12 + 17 > 4\)[/tex]
Let's evaluate each of these conditions:
1. Checking if [tex]\(4 + 12 > 17\)[/tex]:
- The sum is 16, which is not greater than 17.
- Condition 1 is false.
2. Checking if [tex]\(4 + 17 > 12\)[/tex]:
- The sum is 21, which is greater than 12.
- Condition 2 is true.
3. Checking if [tex]\(12 + 17 > 4\)[/tex]:
- The sum is 29, which is greater than 4.
- Condition 3 is true.
For a triangle to exist, all three conditions must be true. Since Condition 1 is false, the given side lengths do not satisfy the triangle inequality theorem, meaning it is not possible to form a triangle with these side lengths.
Therefore, the correct statement is:
This triangle does not exist because the sum of 4 and 12 is less than 17.
