Question

which of the following cannot be the lengths of the three sides of a triangle? 3, 4, 5 inches; 5, 12, 13 cms; 4, 6, 8 yards; 3, 6, 9 mm

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.

Step 1: Check 3, 4, 5 inches

• \( 3 + 4 = 7 \), and \( 7 > 5 \)
• \( 3 + 5 = 8 \), and \( 8 > 4 \)
• \( 4 + 5 = 9 \), and \( 9 > 3 \)

So, 3, 4, 5 inches can form a triangle.

Step 2: Check 5, 12, 13 cms

• \( 5 + 12 = 17 \), and \( 17 > 13 \)
• \( 5 + 13 = 18 \), and \( 18 > 12 \)
• \( 12 + 13 = 25 \), and \( 25 > 5 \)

So, 5, 12, 13 cms can form a triangle.

Step 3: Check 4, 6, 8 yards

• \( 4 + 6 = 10 \), and \( 10 > 8 \)
• \( 4 + 8 = 12 \), and \( 12 > 6 \)
• \( 6 + 8 = 14 \), and \( 14 > 4 \)

So, 4, 6, 8 yards can form a triangle.

Step 4: Check 3, 6, 9 mm

• \( 3 + 6 = 9 \), and \( 9 = 9 \) (not greater than 9)
• \( 3 + 9 = 12 \), and \( 12 > 6 \)
• \( 6 + 9 = 15 \), and \( 15 > 3 \)

Since \( 3 + 6 \) is not greater than 9, 3, 6, 9 mm cannot form a triangle.

Answer:

3, 6, 9 mm