To determine if a triangle can be formed with side lengths of 4 inches, 5 inches, and 8 inches, we need to use the triangle inequality theorem. This theorem states that for any three sides to form a triangle, the following conditions must all be satisfied:
1. The sum of the lengths of any two sides must be greater than the length of the remaining side.
We can break this down into three conditions that must all be satisfied:
1. [tex]\(4 + 5 > 8\)[/tex]
2. [tex]\(4 + 8 > 5\)[/tex]
3. [tex]\(5 + 8 > 4\)[/tex]
Let’s check each condition step-by-step:
1. Check whether [tex]\(4 + 5 > 8\)[/tex]:
[tex]\[
4 + 5 = 9 \quad \text{and} \quad 9 > 8 \quad \text{(True)}
\][/tex]
2. Check whether [tex]\(4 + 8 > 5\)[/tex]:
[tex]\[
4 + 8 = 12 \quad \text{and} \quad 12 > 5 \quad \text{(True)}
\][/tex]
3. Check whether [tex]\(5 + 8 > 4\)[/tex]:
[tex]\[
5 + 8 = 13 \quad \text{and} \quad 13 > 4 \quad \text{(True)}
\][/tex]
Since all three conditions [tex]\((4 + 5 > 8)\)[/tex], [tex]\((4 + 8 > 5)\)[/tex], and [tex]\((5 + 8 > 4)\)[/tex] are met, it confirms that the given side lengths satisfy the triangle inequality theorem. Therefore, a triangle can indeed be formed with side lengths 4 inches, 5 inches, and 8 inches.
So the correct answer is:
Yes, because [tex]\(4 + 5 > 8\)[/tex]
