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.
