Which of the following sets of numbers could represent the three sides of a triangle?

Question

Answer
Multiple Choice Answers
left curly bracket, 4, comma, 12, comma, 18, right curly bracket{4,12,18}
left curly bracket, 5, comma, 20, comma, 25, right curly bracket{5,20,25}
left curly bracket, 7, comma, 13, comma, 21, right curly bracket{7,13,21}
left curly bracket, 5, comma, 9, comma, 12, right curly bracket{5,9,12}

Answer 1

To determine whether a set of three numbers can represent the sides of a triangle, we can use the triangle inequality theorem, which states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):

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

We can check each of the provided sets of numbers against this theorem.

1. Set {4, 12, 18}

   • \( 4 + 12 = 16 \) (not greater than 18)
   • This set does not satisfy the triangle inequality, so it cannot represent the sides of a triangle.

2. Set {5, 20, 25}

   • \( 5 + 20 = 25 \) (not greater than 25)
   • This set does not satisfy the triangle inequality, so it cannot represent the sides of a triangle.

3. Set {7, 13, 21}

   • \( 7 + 13 = 20 \) (not greater than 21)
   • This set does not satisfy the triangle inequality, so it cannot represent the sides of a triangle.

4. Set {5, 9, 12}

   • \( 5 + 9 = 14 > 12 \)
   • \( 5 + 12 = 17 > 9 \)
   • \( 9 + 12 = 21 > 5 \)
   • This set satisfies all triangle inequality conditions, so it can represent the sides of a triangle.

Therefore, the only set of numbers that can represent the sides of a triangle is {5, 9, 12}.