Question

1. which set of numbers represents the lengths of the sides of a triangle? (1) {5, 18, 13} (2) {6, 17, 22} (3) {16, 24, 7} (4) {26, 8, 15}

Explanation:

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

Step 1: Analyze Option (1) {5, 18, 13}

Check the sums:

• \( 5 + 13 = 18 \). But for a triangle, the sum must be greater than the third side, not equal. So \( 5 + 13

ot> 18 \). Thus, this set does not form a triangle.

Step 2: Analyze Option (2) {6, 17, 22}

Check all three sums:

• \( 6 + 17 = 23 \). Since \( 23 > 22 \) (the third side).
• \( 6 + 22 = 28 \). Since \( 28 > 17 \).
• \( 17 + 22 = 39 \). Since \( 39 > 6 \).

All sums satisfy the triangle inequality. Let's verify the other options to be sure.

Step 3: Analyze Option (3) {16, 24, 7}

Check the sum of the two smaller sides:

• \( 7 + 16 = 23 \). Since \( 23 < 24 \) (the largest side), \( 7 + 16

ot> 24 \). Thus, this set does not form a triangle.

Step 4: Analyze Option (4) {26, 8, 15}

Check the sum of the two smaller sides:

• \( 8 + 15 = 23 \). Since \( 23 < 26 \) (the largest side), \( 8 + 15

ot> 26 \). Thus, this set does not form a triangle.

Answer:

(2) {6, 17, 22}