To determine which of the given sets cannot represent the three sides of a triangle, we need to use the triangle inequality theorem. The triangle inequality theorem states that for three sides to form a triangle, the sum of any two sides must be greater than the third side.
Let's analyze each set individually:
1. Set {10, 21, 29}
- Check the sum of two sides against the third:
- \(10 + 21 > 29\) → \(31 > 29\) (True)
- \(10 + 29 > 21\) → \(39 > 21\) (True)
- \(21 + 29 > 10\) → \(50 > 10\) (True)
- All conditions are satisfied, so these sides can form a triangle.
2. Set {9, 20, 28}
- Check the sum of two sides against the third:
- \(9 + 20 > 28\) → \(29 > 28\) (True)
- \(9 + 28 > 20\) → \(37 > 20\) (True)
- \(20 + 28 > 9\) → \(48 > 9\) (True)
- All conditions are satisfied, so these sides can form a triangle.
3. Set {4, 17, 22}
- Check the sum of two sides against the third:
- \(4 + 17 > 22\) → \(21 > 22\) (False)
- \(4 + 22 > 17\) → \(26 > 17\) (True)
- \(17 + 22 > 4\) → \(39 > 4\) (True)
- One condition (\(4 + 17 > 22\)) is not satisfied; thus, these sides cannot form a triangle.
4. Set {11, 20, 29}
- Check the sum of two sides against the third:
- \(11 + 20 > 29\) → \(31 > 29\) (True)
- \(11 + 29 > 20\) → \(40 > 20\) (True)
- \(20 + 29 > 11\) → \(49 > 11\) (True)
- All conditions are satisfied, so these sides can form a triangle.
Hence, the set that cannot represent the three sides of a triangle is:
{4, 17, 22}.
