To determine whether a triangle can be formed with side lengths 4 inches, 5 inches, and 8 inches, we can use the Triangle Inequality Theorem. This theorem states that for any three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's apply the Triangle Inequality Theorem step by step:
1. Check if the sum of the first two sides is greater than the third side:
[tex]\[
4 + 5 > 8
\][/tex]
[tex]\[
9 > 8 \quad \text{(True)}
\][/tex]
2. Check if the sum of the first and third sides is greater than the second side:
[tex]\[
4 + 8 > 5
\][/tex]
[tex]\[
12 > 5 \quad \text{(True)}
\][/tex]
3. Check if the sum of the second and third sides is greater than the first side:
[tex]\[
5 + 8 > 4
\][/tex]
[tex]\[
13 > 4 \quad \text{(True)}
\][/tex]
Since all three conditions of the Triangle Inequality Theorem are satisfied, we can conclude that a triangle can be formed with side lengths 4 inches, 5 inches, and 8 inches.
Therefore, the student is correct, and the correct justification is:
"Yes, because [tex]\(4 + 5 > 8\)[/tex]."
