Question

Which of the following sets of numbers could not represent the three sides of a triangle? Answer Multiple Choice Answers left curly bracket, 4, comma, 16, comma, 17, right curly bracket{4,16,17} left curly bracket, 15, comma, 24, comma, 36, right curly bracket{15,24,36} left curly bracket, 14, comma, 26, comma, 37, right curly bracket{14,26,37} left curly bracket, 12, comma, 24, comma, 37, right curly bracket{12,24,37}

Answer 1

To determine whether a set of three numbers can represent the sides of a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides \(a\), \(b\), and \(c\), the following must be true:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

We will check each given set of numbers against these inequalities.

1. **Set \(\{4, 16, 17\}\)**:
- \(4 + 16 = 20 > 17\) (True)
- \(4 + 17 = 21 > 16\) (True)
- \(16 + 17 = 33 > 4\) (True)
- **This set can represent a triangle.**

2. **Set \(\{15, 24, 36\}\)**:
- \(15 + 24 = 39 > 36\) (True)
- \(15 + 36 = 51 > 24\) (True)
- \(24 + 36 = 60 > 15\) (True)
- **This set can represent a triangle.**

3. **Set \(\{14, 26, 37\}\)**:
- \(14 + 26 = 40 > 37\) (True)
- \(14 + 37 = 51 > 26\) (True)
- \(26 + 37 = 63 > 14\) (True)
- **This set can represent a triangle.**

4. **Set \(\{12, 24, 37\}\)**:
- \(12 + 24 = 36\) (Not greater than 37, so this fails)
- \(12 + 37 = 49 > 24\) (True)
- \(24 + 37 = 61 > 12\) (True)
- **This set cannot represent a triangle.**

Thus, the set of numbers that could not represent the three sides of a triangle is \(\{12, 24, 37\}\).