To determine whether a triangle with side lengths 2 inches, 5 inches, and 4 inches is acute, we need to check the specific properties of the triangle with respect to an acute triangle.
An acute triangle means that all the interior angles are less than 90 degrees. This is equivalent to saying that for any three sides of the triangle, the sum of the squares of the two shorter sides must be greater than the square of the longest side.
Given the side lengths: [tex]\(a = 2\)[/tex] inches, [tex]\(b = 5\)[/tex] inches, and [tex]\(c = 4\)[/tex] inches, let's identify the longest side:
- The longest side is [tex]\(b = 5\)[/tex] inches.
To check if the triangle is acute, we need to compare the squares of the side lengths. The side lengths are [tex]\(2, 4,\)[/tex] and [tex]\(5\)[/tex]. We should determine whether the sum of the squares of the two shorter sides is greater than the square of the longest side. Therefore:
Calculate [tex]\(2^2 + 4^2\)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ 4^2 = 16 \][/tex]
[tex]\[ 2^2 + 4^2 = 4 + 16 = 20 \][/tex]
Compare this to [tex]\(5^2\)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
Now, check if [tex]\(2^2 + 4^2 < 5^2\)[/tex]:
[tex]\[ 20 < 25 \][/tex]
Since the sum of the squares of the shorter sides (20) is less than the square of the longest side (25), we conclude that:
The triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex].
Therefore, the correct explanation is:
The triangle is not acute because [tex]\(2^2 + 4^2 < 5^2\)[/tex].
