`{:("Quantity A","Quantity B"),("The third side of an isosceles","The third side of an isoscles"),("triangle with sides of 3 and 9","triangle with sides of 6 and 8"):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the problem, we need to determine the third sides of two isosceles triangles based on the given side lengths.

### Step-by-Step Solution:

**Step 1: Analyze Quantity A**
- Quantity A is the third side of an isosceles triangle with sides of 3 and 9.
- In an isosceles triangle, two sides are equal. Therefore, the third side can either be equal to 3 or equal to 9.

**Step 2: Check if the third side can be 3**
- If we assume the third side is 3, we check the triangle inequality:
- The sum of the two equal sides (3 + 3 = 6) must be greater than the third side (9).
- Here, 6 is not greater than 9, so this configuration is not valid.

**Step 3: Conclude for Quantity A**
- Since the third side cannot be 3, the only possibility left is that the third side is 9.
- Thus, Quantity A = 9.

**Step 4: Analyze Quantity B**
- Quantity B is the third side of an isosceles triangle with sides of 6 and 8.
- Again, the third side can either be equal to 6 or equal to 8.

**Step 5: Check if the third side can be 6**
- If we assume the third side is 6, we check the triangle inequality:
- The sum of the two equal sides (6 + 8 = 14) must be greater than the third side (6).
- Here, 14 is greater than 6, so this configuration is valid.

**Step 6: Check if the third side can be 8**
- If we assume the third side is 8, we check the triangle inequality:
- The sum of the two equal sides (6 + 8 = 14) must be greater than the third side (8).
- Here, 14 is greater than 8, so this configuration is also valid.

**Step 7: Conclude for Quantity B**
- The third side can be either 6 or 8.
- Thus, Quantity B = 6 or 8.

**Step 8: Compare Quantities**
- Quantity A = 9
- Quantity B can be 6 or 8.
- Therefore, Quantity A (9) is greater than Quantity B (6 or 8).

### Final Conclusion:
- Quantity A is greater than Quantity B.