To determine which set of side lengths can form the side lengths of an acute triangle, we need to examine each set and apply the criteria for acute triangles. A triangle is acute if all its angles are less than 90 degrees. For this to be true, 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. Mathematically, for a triangle with sides [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex]:
[tex]\[ a^2 + b^2 > c^2 \][/tex]
[tex]\[ a^2 + c^2 > b^2 \][/tex]
[tex]\[ b^2 + c^2 > a^2 \][/tex]
We will check each set of side lengths:
1. Set: [tex]\( (4, 5, 7) \)[/tex]
- Check [tex]\( 4^2 + 5^2 > 7^2 \)[/tex]:
[tex]\[ 16 + 25 > 49 \][/tex]
[tex]\[ 41 \not> 49 \][/tex]
- Since [tex]\( 4^2 + 5^2 \not> 7^2 \)[/tex], this set does not form an acute triangle.
2. Set: [tex]\( (5, 7, 8) \)[/tex]
- Check [tex]\( 5^2 + 7^2 > 8^2 \)[/tex]:
[tex]\[ 25 + 49 > 64 \][/tex]
[tex]\[ 74 > 64 \][/tex] (true)
- Check [tex]\( 5^2 + 8^2 > 7^2 \)[/tex]:
[tex]\[ 25 + 64 > 49 \][/tex]
[tex]\[ 89 > 49 \][/tex] (true)
- Check [tex]\( 7^2 + 8^2 > 5^2 \)[/tex]:
[tex]\[ 49 + 64 > 25 \][/tex]
[tex]\[ 113 > 25 \][/tex] (true)
- All conditions are met, so [tex]\( (5, 7, 8) \)[/tex] can form an acute triangle.
3. Set: [tex]\( (6, 7, 10) \)[/tex]
- Check [tex]\( 6^2 + 7^2 > 10^2 \)[/tex]:
[tex]\[ 36 + 49 > 100 \][/tex]
[tex]\[ 85 \not> 100 \][/tex]
- Since [tex]\( 6^2 + 7^2 \not> 10^2 \)[/tex], this set does not form an acute triangle.
4. Set: [tex]\( (7, 9, 12) \)[/tex]
- Check [tex]\( 7^2 + 9^2 > 12^2 \)[/tex]:
[tex]\[ 49 + 81 > 144 \][/tex]
[tex]\[ 130 \not> 144 \][/tex]
- Since [tex]\( 7^2 + 9^2 \not> 12^2 \)[/tex], this set does not form an acute triangle.
Thus, the set of numbers that can represent the side lengths, in inches, of an acute triangle is:
[tex]\[ (5, 7, 8) \][/tex]
