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A page in Johanie's workbook is torn, and one of the questions is cut off. She can read only the first part of the question:

"In which quadrant would you find point $P$ if the coordinates of $P$ are $(-5, \ldots$ "



Assuming that $P$ is not on an axis, what are the possible answers to this question?


Aquadrant I or quadrant II
Bquadrant II or quadrant III
Cquadrant II or quadrant IV
Dquadrant III or quadrant IV

Verified Answer :

  • Analyze the x-coordinate of the point \(P(-5, y)\), which is negative (\(x = -5\)).
  • Determine that quadrants with a negative x-coordinate are Quadrant II and Quadrant III.
  • Conclude that the point must lie in either Quadrant II or Quadrant III.
  • The final answer is \(\boxed{\text{quadrant II or quadrant III}}\).

Explanation

  1. Analyze the coordinates and identify possible quadrants.

The point \(P\) is given with coordinates \((-5, y)\). We are given that \(P\) is not on an axis, which means that neither \(x\) nor \(y\) is equal to 0. Since the x-coordinate is \(-5\), which is negative, we need to identify the quadrants where the x-coordinate is negative. In the Cartesian coordinate system:

  • Quadrant I: \(x > 0, y > 0\)
  • Quadrant II: \(x < 0, y > 0\)
  • Quadrant III: \(x < 0, y < 0\)
  • Quadrant IV: \(x > 0, y < 0\)

Since the x-coordinate is negative (\(-5\)), the point must lie in a quadrant where \(x\) is negative. Looking at the definitions above, the quadrants with a negative x-coordinate are Quadrant II and Quadrant III. Therefore, regardless of whether \(y\) is positive or negative, the point \(P\) will always be in either Quadrant II or Quadrant III.

  1. Select the correct option based on the analysis.

Based on our analysis, the possible quadrants for point \(P(-5, y)\) are Quadrant II or Quadrant III. Comparing this with the provided options:

  • quadrant I or quadrant II
  • quadrant II or quadrant III
  • quadrant II or quadrant IV
  • quadrant III or quadrant IV

The correct option is "quadrant II or quadrant III". The final answer is \(\boxed{\text{quadrant II or quadrant III}}\).

Examples

In geography, coordinate systems are used to map locations. If you are told a location has a negative longitude (west of the Prime Meridian), you immediately know it must be in the western hemisphere, which corresponds to the left half of a coordinate map. Similarly, knowing a specific coordinate value restricts the possible regions where a point can exist, just as knowing that x = -5 restricts the point to the left side of the y-axis, narrowing down its possible location to either Quadrant II or Quadrant III.

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