To determine which of the given sets of numbers cannot represent the three sides of a triangle, we need to use the triangle inequality theorem. This theorem states that for any three sides \(a\), \(b\), and \(c\) to form a triangle, the following conditions must all be true:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Let's check each set of numbers:
1. \(\{7, 10, 15\}\)
- Check \(7 + 10 > 15\): \(17 > 15\) (True)
- Check \(7 + 15 > 10\): \(22 > 10\) (True)
- Check \(10 + 15 > 7\): \(25 > 7\) (True)
All conditions are satisfied, so this set can form a triangle.
2. \(\{5, 15, 20\}\)
- Check \(5 + 15 > 20\): \(20 \nless 20\) (False)
- Since this condition is false, the other conditions do not need to be checked.
This set does not satisfy the conditions of the triangle inequality theorem, so these numbers cannot form a triangle.
3. \(\{6, 15, 19\}\)
- Check \(6 + 15 > 19\): \(21 > 19\) (True)
- Check \(6 + 19 > 15\): \(25 > 15\) (True)
- Check \(15 + 19 > 6\): \(34 > 6\) (True)
All conditions are satisfied, so this set can form a triangle.
4. \(\{4, 12, 15\}\)
- Check \(4 + 12 > 15\): \(16 > 15\) (True)
- Check \(4 + 15 > 12\): \(19 > 12\) (True)
- Check \(12 + 15 > 4\): \(27 > 4\) (True)
All conditions are satisfied, so this set can form a triangle.
From this analysis, the set of numbers \(\{5, 15, 20\}\) cannot represent the three sides of a triangle.
So, the correct answer is:
○ {5, 15, 20}
