Question

Which of the following sets of numbers could represent the three sides of a triangle?
Answer
Multiple Choice Answers
left curly bracket, 15, comma, 25, comma, 39, right curly bracket{15,25,39}
left curly bracket, 5, comma, 9, comma, 16, right curly bracket{5,9,16}
left curly bracket, 5, comma, 14, comma, 19, right curly bracket{5,14,19}
left curly bracket, 9, comma, 23, comma, 33, right curly bracket{9,23,33}

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 three sides \( a \), \( b \), and \( c \) of a triangle, the following conditions must be met:

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

Let’s evaluate each set of numbers:

1. **Set {15, 25, 39}**
- \( 15 + 25 = 40 > 39 \) (True)
- \( 15 + 39 = 54 > 25 \) (True)
- \( 25 + 39 = 64 > 15 \) (True)
- All conditions are satisfied, so this set **can form a triangle**.

2. **Set {5, 9, 16}**
- \( 5 + 9 = 14 < 16 \) (False)
- Since the first condition fails, this set **cannot form a triangle**.

3. **Set {5, 14, 19}**
- \( 5 + 14 = 19 \) (not greater than)
- Since the first condition fails, this set **cannot form a triangle**.

4. **Set {9, 23, 33}**
- \( 9 + 23 = 32 < 33 \) (False)
- Since the first condition fails, this set **cannot form a triangle**.

In conclusion, the only set that can represent the sides of a triangle is:

**{15, 25, 39}**.