To determine whether the given triangle with side lengths 2 inches, 5 inches, and 4 inches is acute, we need to check the properties of the triangle according to the sides' relationship.
For a triangle to be acute, the square of the length of each side must be less than the sum of the squares of the lengths of the other two sides. This is part of the general triangle inequality theorem extended for acute triangles.
1. The given sides are 2 inches, 5 inches, and 4 inches.
2. First, we check the square of each side:
[tex]\[
2^2 = 4, \quad 5^2 = 25, \quad 4^2 = 16
\][/tex]
3. Now, we compare the square of the longest side (25) with the sum of the squares of the other two sides (4 and 16):
[tex]\[
2^2 + 4^2 = 4 + 16 = 20
\][/tex]
4. Since [tex]\(2^2 + 4^2 = 20\)[/tex] is less than [tex]\(5^2 = 25\)[/tex], we know that:
[tex]\[
20 < 25
\][/tex]
5. Thus, since the sum of the squares of the two shorter sides (20) is less than the square of the longest side (25), the triangle is not an acute triangle.
Hence, the correct explanation is:
The triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex].
