Final answer:
The set of measurements that cannot represent the three side lengths of a triangle is D. 3 cm, 6 cm, 9 cm because it violates the Triangle Inequality Theorem.
Step-by-step explanation:
The question asks which set of measurements cannot represent the three side lengths of a triangle. To determine this, we can 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 third side. Let's apply this to each set:
- A. 3 cm, 3 cm, 3 cm: This set satisfies the theorem since 3+3 > 3, and it represents an equilateral triangle.
- B. 3 cm, 4 cm, 5 cm: This also satisfies the theorem (3+4 > 5, 3+5 > 4, 4+5 > 3) and represents a right triangle.
- C. 3 cm, 5 cm, 7 cm: This set barely satisfies the theorem (3+5 > 7, 5+7 > 3, 3+7 > 5), so it can represent a triangle.
- D. 3 cm, 6 cm, 9 cm: Here, the sum of the two smaller sides equals the largest side (3+6 = 9), which does not satisfy the theorem. Therefore, these measurements cannot form a triangle.
Therefore, the set of measurements that cannot represent the three side lengths of a triangle is D. 3 cm, 6 cm, 9 cm.