Answer :
To determine whether a triangle with side lengths 2 inches, 5 inches, and 4 inches is acute, we need to analyze the given answer step by step.
First, let's recall the properties of triangles:
1. In an acute triangle, all angles are less than 90 degrees.
2. For a triangle to be formed, the sum of any two side lengths must be greater than the third side.
Next, let's square each of the side lengths:
- The first side squared: [tex]\(2^2 = 4\)[/tex]
- The second side squared: [tex]\(5^2 = 25\)[/tex]
- The third side squared: [tex]\(4^2 = 16\)[/tex]
Now, let's check if the triangle is acute by examining the conditions for its angles. For a triangle to be acute, the sum of the squares of any two side lengths must be greater than the square of the third side.
### Check:
1. [tex]\(2^2 + 4^2\)[/tex]
[tex]\[ 2^2 + 4^2 = 4 + 16 = 20 \][/tex]
Since [tex]\(20 < 5^2\)[/tex]:
[tex]\[ 20 < 25 \][/tex]
This indicates that the sum of the squares of the smaller two sides is less than the square of the largest side (5).
Based on this inequality, we can conclude that the triangle is not acute because the sum of the squares of two sides (2 and 4) is less than the square of the longest side (5).
So, the most accurate explanation is:
[tex]\[ \text{The triangle is not acute because } 2^2 + 4^2 < 5^2. \][/tex]
First, let's recall the properties of triangles:
1. In an acute triangle, all angles are less than 90 degrees.
2. For a triangle to be formed, the sum of any two side lengths must be greater than the third side.
Next, let's square each of the side lengths:
- The first side squared: [tex]\(2^2 = 4\)[/tex]
- The second side squared: [tex]\(5^2 = 25\)[/tex]
- The third side squared: [tex]\(4^2 = 16\)[/tex]
Now, let's check if the triangle is acute by examining the conditions for its angles. For a triangle to be acute, the sum of the squares of any two side lengths must be greater than the square of the third side.
### Check:
1. [tex]\(2^2 + 4^2\)[/tex]
[tex]\[ 2^2 + 4^2 = 4 + 16 = 20 \][/tex]
Since [tex]\(20 < 5^2\)[/tex]:
[tex]\[ 20 < 25 \][/tex]
This indicates that the sum of the squares of the smaller two sides is less than the square of the largest side (5).
Based on this inequality, we can conclude that the triangle is not acute because the sum of the squares of two sides (2 and 4) is less than the square of the longest side (5).
So, the most accurate explanation is:
[tex]\[ \text{The triangle is not acute because } 2^2 + 4^2 < 5^2. \][/tex]
