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

Question

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}.