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The problem is asking whether the ratio \(\frac{x}{r}\) is positive or negative, where \(r\) is a positive number and the point \((x, y)\) is located in the third quadrant (Quadrant III).

### Key points:
1. In Quadrant III:
   - \(x\) is **negative** (since it's to the left of the origin).
   - \(y\) is also **negative** (since it's below the origin).

2. \(r\) is the distance from the origin to the point \((x, y)\), and by definition, \(r\) is positive because it represents the radius or distance.

### Now, analyzing \(\frac{x}{r}\):
- \(x\) is negative (from Quadrant III).
- \(r\) is positive (as given).

Therefore, the ratio \(\frac{x}{r}\) is **negative** because dividing a negative number by a positive number results in a negative value.

### Conclusion:
The given ratio \(\frac{x}{r}\) is **negative**.

Let me know if you want more details or have any questions!

Here are five related questions to expand on this:
1. What is the sign of \(\frac{y}{r}\) in Quadrant III?
2. How does the sign of \(\frac{x}{r}\) change in other quadrants?
3. What would the ratio \(\frac{y}{x}\) be in Quadrant III?
4. Why is \(r\) always positive in polar coordinates?
5. How does the Pythagorean theorem relate to finding \(r\)?

**Tip**: In any quadrant, remember the signs of \(x\) and \(y\) to quickly determine the signs of related ratios.

If r is a positive number and the point (x, y) is in the indicated quadrant, decide whether the given ratio x/r is positive or negative. (Quadrant III)

Explore how to determine whether the ratio x/r is positive or negative for points located in Quadrant III. Learn about key concepts in coordinate geometry and signs of coordinates, suitable for students in Grades 9-11.

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