To determine whether a triangle with side lengths 2 inches, 5 inches, and 4 inches is an acute triangle, we need to consider two main criteria:
1. The triangle inequality theorem.
2. The nature of the angles formed.
### Step 1: Checking 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. This ensures that the given side lengths can form a triangle.
For the side lengths 2, 5, and 4:
- Check [tex]\(2 + 5 > 4\)[/tex]. This is true because [tex]\(7 > 4\)[/tex].
- Check [tex]\(2 + 4 > 5\)[/tex]. This is true because [tex]\(6 > 5\)[/tex].
- Check [tex]\(5 + 4 > 2\)[/tex]. This is true because [tex]\(9 > 2\)[/tex].
Since all three conditions are satisfied, the side lengths can form a triangle.
### Step 2: Determining if the Triangle is Acute
An acute triangle is one where all three angles are less than 90 degrees. We can use the relationship between the squares of the side lengths to determine the nature of the angles. Specifically, we compare the sum of the squares of the two shorter sides to the square of the longest side.
Let's consider the given side lengths:
- [tex]\(a = 2\)[/tex] inches
- [tex]\(b = 5\)[/tex] inches
- [tex]\(c = 4\)[/tex] inches
We compute the squares of the side lengths:
- [tex]\(a^2 = 2^2 = 4\)[/tex]
- [tex]\(b^2 = 5^2 = 25\)[/tex]
- [tex]\(c^2 = 4^2 = 16\)[/tex]
We now compare [tex]\(a^2 + c^2\)[/tex] with [tex]\(b^2\)[/tex]:
- Compute [tex]\(a^2 + c^2\)[/tex]: [tex]\(4 + 16 = 20\)[/tex]
- Compare [tex]\(a^2 + c^2\)[/tex] with [tex]\(b^2\)[/tex]: [tex]\(20 < 25\)[/tex].
Since [tex]\(a^2 + c^2 < b^2\)[/tex], it indicates that the triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex].
Therefore, the best explanation is:
The triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex].
This matches the answer:
"The triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex]."
