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.

Step1: 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.

Step2: 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.

Step3: 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.

Step4: Check 3, 6, 9 mm

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

Since the sum of 3 and 6 is not greater than 9, these lengths do not satisfy the triangle inequality theorem.

Answer:

3, 6, 9 mm