Which sets of numbers can represent the lengths of the sides of a triangle?

To determine whether a set of numbers can represent the lengths of the sides of a triangle, we apply the triangle inequality theorem. This theorem states that for any triangle with sides of lengths a, b, and c, the following conditions must hold:

1. a + b > c

2. a + c > b

3. b + c > a

This means the sum of the lengths of any two sides must be greater than the length of the third side.

When evaluating each set of numbers, if all combinations satisfy this theorem, then those sets can represent the lengths of a triangle.

The correct answer implies that all specified sets (A, B, and D) meet the triangle inequality criteria. When checking these sets, you would find that they all pass the tests laid out by the triangle inequality theorem.

This reinforces the importance of the triangle inequality theorem in determining valid lengths for the sides of a triangle and shows why selecting all qualifying sets (in this case, A, B, and D) is necessary for a comprehensive understanding of triangle side lengths.