Question

select all of the sets of numbers that could be the side lengths of a triangle
a. 7, 13, 21
b. 6, 6, 2
c. 15, 17, 34
d. 9, 10, 11
e. 4, 8, 12

Explanation:

To determine if a set of numbers can be the side lengths of a triangle, we use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side. We will check each option:

Option A: \( 7, 13, 21 \)

• Check the sums:
• \( 7 + 13 = 20 \). Since \( 20 < 21 \), this does not satisfy the triangle inequality theorem. So, this set cannot be the side lengths of a triangle.

Option B: \( 6, 6, 2 \)

• Check the sums:
• \( 6 + 6 = 12 \), and \( 12 > 2 \)
• \( 6 + 2 = 8 \), and \( 8 > 6 \)
• \( 6 + 2 = 8 \), and \( 8 > 6 \)

All sums satisfy the triangle inequality theorem. So, this set can be the side lengths of a triangle.

Option C: \( 15, 17, 34 \)

• Check the sums:
• \( 15 + 17 = 32 \). Since \( 32 < 34 \), this does not satisfy the triangle inequality theorem. So, this set cannot be the side lengths of a triangle.

Option D: \( 9, 10, 11 \)

• Check the sums:
• \( 9 + 10 = 19 \), and \( 19 > 11 \)
• \( 9 + 11 = 20 \), and \( 20 > 10 \)
• \( 10 + 11 = 21 \), and \( 21 > 9 \)

All sums satisfy the triangle inequality theorem. So, this set can be the side lengths of a triangle.

Option E: \( 4, 8, 12 \)

• Check the sums:
• \( 4 + 8 = 12 \). Since \( 12 = 12 \), this does not satisfy the triangle inequality theorem (needs to be greater, not equal). So, this set cannot be the side lengths of a triangle.

The sets that could be the side lengths of a triangle are B and D.

Answer:

B. 6, 6, 2
D. 9, 10, 11