Question
In which quadrant, if any, is the point \( \left(\frac{4}{9}, \frac{20}{3}\right) \) located?
Ask by Greene Wilson. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The point \( \left(\frac{4}{9}, \frac{20}{3}\right) \) is in Quadrant I.
Solution
To determine in which quadrant the point \( \left(\frac{4}{9}, \frac{20}{3}\right) \) is located, let's analyze the signs of its \( x \)- and \( y \)-coordinates.
1. **\( x \)-coordinate (\(\frac{4}{9}\))**:
- \( \frac{4}{9} > 0 \) (positive)
2. **\( y \)-coordinate (\(\frac{20}{3}\))**:
- \( \frac{20}{3} > 0 \) (positive)
In the Cartesian coordinate system, the plane is divided into four quadrants based on the signs of the \( x \)- and \( y \)-coordinates:
- **Quadrant I**: \( x > 0 \) and \( y > 0 \)
- **Quadrant II**: \( x < 0 \) and \( y > 0 \)
- **Quadrant III**: \( x < 0 \) and \( y < 0 \)
- **Quadrant IV**: \( x > 0 \) and \( y < 0 \)
Since both coordinates of the point \( \left(\frac{4}{9}, \frac{20}{3}\right) \) are positive, the point lies in **Quadrant I**.
**Answer:** Quadrant I
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Simplify this solution The Deep Dive
To determine the quadrant of the point \( \left(\frac{4}{9}, \frac{20}{3}\right) \), we need to evaluate the signs of the x-coordinate and y-coordinate. - The x-coordinate is \( \frac{4}{9} \), which is positive. - The y-coordinate is \( \frac{20}{3} \), which is also positive. In the Cartesian coordinate system: - Quadrant I has both coordinates positive. - Quadrant II has a negative x-coordinate and a positive y-coordinate. - Quadrant III has both coordinates negative. - Quadrant IV has a positive x-coordinate and a negative y-coordinate. Since both coordinates of the point \( \left(\frac{4}{9}, \frac{20}{3}\right) \) are positive, the point is located in Quadrant I.
