Question

for each set of three lengths, determine if they can be the side lengths of a triangle.

lengths	can be side lengths of a triangle	cannot be side lengths of a triangle
12, 14, 13	○	○
9.8, 5.4, 5.3	○	○
12, 16, 4	○	○

Explanation:

To determine if three lengths can form 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'll apply this to each set of lengths.

Step 1: Analyze 8, 16, 3

Check the sums:

• \( 8 + 3 = 11 \), and \( 11 < 16 \). Since the sum of two sides is not greater than the third, these cannot form a triangle.

Step 2: Analyze 12, 14, 13

Check the sums:

• \( 12 + 14 = 26 > 13 \)
• \( 12 + 13 = 25 > 14 \)
• \( 14 + 13 = 27 > 12 \)

All sums satisfy the inequality, so these can form a triangle.

Step 3: Analyze 9.8, 5.4, 5.3

Check the sums:

• \( 5.4 + 5.3 = 10.7 > 9.8 \)
• \( 9.8 + 5.3 = 15.1 > 5.4 \)
• \( 9.8 + 5.4 = 15.2 > 5.3 \)

All sums satisfy the inequality, so these can form a triangle.

Step 4: Analyze 12, 16, 4

Check the sums:

• \( 12 + 4 = 16 \), and \( 16 = 16 \) (not greater than). Since the sum of two sides is not greater than the third, these cannot form a triangle.

Answer:

• For 8, 16, 3: Cannot be side lengths of a triangle (select the "Cannot be side lengths of a triangle" option).
• For 12, 14, 13: Can be side lengths of a triangle (select the "Can be side lengths of a triangle" option).
• For 9.8, 5.4, 5.3: Can be side lengths of a triangle (select the "Can be side lengths of a triangle" option).
• For 12, 16, 4: Cannot be side lengths of a triangle (select the "Cannot be side lengths of a triangle" option).