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To determine which set of numbers can represent the side lengths of an obtuse triangle, we need to understand the properties of an obtuse triangle. An obtuse triangle is a triangle in which one of the angles is greater than 90 degrees.
For any triangle, if we label the side lengths as [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] where [tex]\(c\)[/tex] is the longest side, the triangle is obtuse if and only if [tex]\(a^2 + b^2 < c^2\)[/tex].
Let's test each set of side lengths:
1. Set: [tex]\(8, 10, 14\)[/tex]
- Identify the longest side: [tex]\(c = 14\)[/tex]
- Check the condition for obtuse triangle: [tex]\(8^2 + 10^2 < 14^2\)[/tex]
[tex]\[ 8^2 = 64 \\ 10^2 = 100 \\ 14^2 = 196 \\ 64 + 100 = 164 \\ 164 < 196 \][/tex]
Since 164 is less than 196, the condition is satisfied. Therefore, the set [tex]\(8, 10, 14\)[/tex] can form an obtuse triangle.
2. Set: [tex]\(9, 12, 15\)[/tex]
- Identify the longest side: [tex]\(c = 15\)[/tex]
- Check the condition for obtuse triangle: [tex]\(9^2 + 12^2 < 15^2\)[/tex]
[tex]\[ 9^2 = 81 \\ 12^2 = 144 \\ 15^2 = 225 \\ 81 + 144 = 225 \\ 225 = 225 \][/tex]
Since 225 is not less than 225, the condition is not satisfied. Therefore, the set [tex]\(9, 12, 15\)[/tex] cannot form an obtuse triangle.
3. Set: [tex]\(10, 14, 17\)[/tex]
- Identify the longest side: [tex]\(c = 17\)[/tex]
- Check the condition for obtuse triangle: [tex]\(10^2 + 14^2 < 17^2\)[/tex]
[tex]\[ 10^2 = 100 \\ 14^2 = 196 \\ 17^2 = 289 \\ 100 + 196 = 296 \\ 296 > 289 \][/tex]
Since 296 is greater than 289, the condition is not satisfied. Therefore, the set [tex]\(10, 14, 17\)[/tex] cannot form an obtuse triangle.
4. Set: [tex]\(12, 15, 19\)[/tex]
- Identify the longest side: [tex]\(c = 19\)[/tex]
- Check the condition for obtuse triangle: [tex]\(12^2 + 15^2 < 19^2\)[/tex]
[tex]\[ 12^2 = 144 \\ 15^2 = 225 \\ 19^2 = 361 \\ 144 + 225 = 369 \\ 369 > 361 \][/tex]
Since 369 is greater than 361, the condition is not satisfied. Therefore, the set [tex]\(12, 15, 19\)[/tex] cannot form an obtuse triangle.
After testing each set, the set [tex]\(8, 10, 14\)[/tex] is the only one that can represent the side lengths of an obtuse triangle.
For any triangle, if we label the side lengths as [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] where [tex]\(c\)[/tex] is the longest side, the triangle is obtuse if and only if [tex]\(a^2 + b^2 < c^2\)[/tex].
Let's test each set of side lengths:
1. Set: [tex]\(8, 10, 14\)[/tex]
- Identify the longest side: [tex]\(c = 14\)[/tex]
- Check the condition for obtuse triangle: [tex]\(8^2 + 10^2 < 14^2\)[/tex]
[tex]\[ 8^2 = 64 \\ 10^2 = 100 \\ 14^2 = 196 \\ 64 + 100 = 164 \\ 164 < 196 \][/tex]
Since 164 is less than 196, the condition is satisfied. Therefore, the set [tex]\(8, 10, 14\)[/tex] can form an obtuse triangle.
2. Set: [tex]\(9, 12, 15\)[/tex]
- Identify the longest side: [tex]\(c = 15\)[/tex]
- Check the condition for obtuse triangle: [tex]\(9^2 + 12^2 < 15^2\)[/tex]
[tex]\[ 9^2 = 81 \\ 12^2 = 144 \\ 15^2 = 225 \\ 81 + 144 = 225 \\ 225 = 225 \][/tex]
Since 225 is not less than 225, the condition is not satisfied. Therefore, the set [tex]\(9, 12, 15\)[/tex] cannot form an obtuse triangle.
3. Set: [tex]\(10, 14, 17\)[/tex]
- Identify the longest side: [tex]\(c = 17\)[/tex]
- Check the condition for obtuse triangle: [tex]\(10^2 + 14^2 < 17^2\)[/tex]
[tex]\[ 10^2 = 100 \\ 14^2 = 196 \\ 17^2 = 289 \\ 100 + 196 = 296 \\ 296 > 289 \][/tex]
Since 296 is greater than 289, the condition is not satisfied. Therefore, the set [tex]\(10, 14, 17\)[/tex] cannot form an obtuse triangle.
4. Set: [tex]\(12, 15, 19\)[/tex]
- Identify the longest side: [tex]\(c = 19\)[/tex]
- Check the condition for obtuse triangle: [tex]\(12^2 + 15^2 < 19^2\)[/tex]
[tex]\[ 12^2 = 144 \\ 15^2 = 225 \\ 19^2 = 361 \\ 144 + 225 = 369 \\ 369 > 361 \][/tex]
Since 369 is greater than 361, the condition is not satisfied. Therefore, the set [tex]\(12, 15, 19\)[/tex] cannot form an obtuse triangle.
After testing each set, the set [tex]\(8, 10, 14\)[/tex] is the only one that can represent the side lengths of an obtuse triangle.
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