Question

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

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), the sum of the lengths of any two sides must be greater than the length of the remaining side. Mathematically, this means:

• \(a + b > c\)
• \(a + c > b\)
• \(b + c > a\)

Step2: Check Option 1 (3 inches, 4 inches, 5 inches)

Let \(a = 3\), \(b = 4\), \(c = 5\).

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

All inequalities hold, so this is a valid triangle.

Step3: Check Option 2 (5 cm, 12 cm, 13 cm)

Let \(a = 5\), \(b = 12\), \(c = 13\).

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

All inequalities hold, so this is a valid triangle.

Step4: Check Option 3 (4 yards, 6 yards, 8 yards)

Let \(a = 4\), \(b = 6\), \(c = 8\).

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

All inequalities hold, so this is a valid triangle.

Step5: Check Option 4 (3 mm, 6 mm, 9 mm)

Let \(a = 3\), \(b = 6\), \(c = 9\).

• \(3 + 6 = 9\), but the triangle inequality requires the sum to be greater than the third side, not equal. So \(3 + 6

ot> 9\).
This violates the triangle inequality theorem.

Answer:

3 millimeters, 6 millimeters, 9 millimeters