Coordinate geometry (or analytic geometry) bridges algebra and geometry by describing the position of points using numbers called coordinates.
- The Cartesian Plane: Two perpendicular number lines — the horizontal x-axis and the vertical y-axis — intersect at the origin O(0, 0). They divide the plane into four quadrants:
- Quadrant I: x > 0, y > 0
- Quadrant II: x < 0, y > 0
- Quadrant III: x < 0, y < 0
- Quadrant IV: x > 0, y < 0
A point P is represented as an ordered pair (x, y) where x is the abscissa (distance from y-axis) and y is the ordinate (distance from x-axis).
Plot the point A(3, -4).
Move 3 units right along x-axis, then 4 units down (negative y). The point lies in Quadrant IV.
In which quadrant does (-5, 2) lie?
x = -5 (negative), y = 2 (positive). This is Quadrant II.
Find the point that lies on both axes simultaneously.
A point on the x-axis has y = 0. A point on the y-axis has x = 0. Both conditions give (0, 0), the origin.
The point (3, 0) lies on which axis?
Since y = 0, the point lies on the x-axis.
What is the distance of the point (-3, 4) from the x-axis?
The distance from the x-axis is the absolute value of the ordinate = |4| = 4 units.
Mirror image of (3, 5) in the x-axis.
When reflecting in the x-axis, the y-coordinate changes sign. Mirror image = (3, -5).
Mirror image of (2, -3) in the y-axis.
When reflecting in the y-axis, the x-coordinate changes sign. Mirror image = (-2, -3).
Summary
- Points on the x-axis: (a, 0) for any a.
- Points on the y-axis: (0, b) for any b.
- Distance from y-axis = |x|; distance from x-axis = |y|.
Common mistakes
Students confuse abscissa and ordinate — remember the x-coordinate (left-right) comes first. Also, do not confuse the quadrant of a point on an axis — axis points are not in any quadrant.
Summary
Every point in a plane is uniquely identified by an ordered pair (x, y). The four quadrants have specific sign patterns for x and y. The Cartesian coordinate system is the backbone of graphing functions and geometric figures.