Question

Which of the following sets of numbers could represent the three sides of a triangle?
Answer
Multiple Choice Answers
left curly bracket, 12, comma, 15, comma, 28, right curly bracket{12,15,28}
left curly bracket, 11, comma, 21, comma, 32, right curly bracket{11,21,32}
left curly bracket, 12, comma, 22, comma, 33, right curly bracket{12,22,33}
left curly bracket, 12, comma, 24, comma, 36, right curly bracket{12,24,36}

Answer 1

To determine whether a set of numbers can represent the sides of a triangle, we can apply the triangle inequality theorem. This theorem states that for any three sides \( a \), \( b \), and \( c \) (where \( a \leq b \leq c \)), the following conditions must hold:

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

However, the most commonly used inequality to check is \( a + b > c \) when the sides are ordered.

Let's analyze each of the given sets:

1. **Set {12, 15, 28}**
- Ordered: \( 12, 15, 28 \)
- Check: \( 12 + 15 = 27 \) which is **not greater than** 28.
- This set **cannot** represent the sides of a triangle.

2. **Set {11, 21, 32}**
- Ordered: \( 11, 21, 32 \)
- Check: \( 11 + 21 = 32 \) which is **not greater than** 32.
- This set **cannot** represent the sides of a triangle.

3. **Set {12, 22, 33}**
- Ordered: \( 12, 22, 33 \)
- Check: \( 12 + 22 = 34 \) which is **greater than** 33.
- Check: \( 12 + 33 = 45 \) which is **greater than** 22.
- Check: \( 22 + 33 = 55 \) which is **greater than** 12.
- This set **can** represent the sides of a triangle.

4. **Set {12, 24, 36}**
- Ordered: \( 12, 24, 36 \)
- Check: \( 12 + 24 = 36 \) which is **not greater than** 36.
- This set **cannot** represent the sides of a triangle.

Thus, the only set that can represent the sides of a triangle is **{12, 22, 33}**.