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Coordinate Geometry

The Cartesian Plane

What is the Cartesian Plane?

The Cartesian plane (or coordinate plane) is a two-dimensional plane formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Their intersection point is called the origin, denoted by O(0, 0).

What are Coordinates of a Point?

A point's coordinates are an ordered pair (x, y) that tells its exact location on the plane:

  • The x-coordinate (abscissa) tells how far to move left or right from the origin
  • The y-coordinate (ordinate) tells how far to move up or down from the origin

How to Plot a Point (x, y):

  1. Start at the origin (0, 0)
  2. Move horizontally: x units right (if x > 0) or left (if x < 0)
  3. Move vertically: y units up (if y > 0) or down (if y < 0)
  4. Mark the point where you end

Real-Life Applications:

  • Maps and GPS navigation (latitude and longitude)
  • Video game character positioning
  • Architectural blueprints and floor plans
  • Air traffic control radar screens
The Coordinate PlanexyO-3-2-1123Q I (+,+)Q II (−,+)Q III (−,−)Q IV (+,−)P(2, 2)22
Example 1: In which quadrant does (−3, 5) lie?
x is negative and y positive, so it is in Quadrant II.
Example 2: Where does (0, −4) lie?
Its x-coordinate is 0, so it lies on the y-axis (no quadrant).
Quick recap
  • Axes meet at the origin (0, 0); quadrants are numbered anticlockwise.
  • Signs (±, ±) fix the quadrant; points on an axis are in none.
✓ Quick check
Find the distance between P(−3, 2) and Q(5, 2).
Same y-coordinate, so the distance is |5 − (−3)| = 8 units.
If A(x, 4) lies on a line parallel to the y-axis passing through B(5, −2), then x is:
A line parallel to the y-axis has a constant x; through B(5, −2) it is x = 5, so x = 5.

Plotting Points: Abscissa and Ordinate

How to Plot Points Systematically:

Plotting means marking a point on the Cartesian plane given its coordinates (x, y).

Step-by-Step Plotting Process:

  1. Locate the origin (0, 0)
  2. Move horizontally along the x-axis to the x-coordinate
  3. From that position, move vertically to the y-coordinate
  4. Mark and label the point

Plotting Points with Same x or Same y:

  • Points with same x-coordinate lie on a vertical line
  • Points with same y-coordinate lie on a horizontal line

Plotting Multiple Points:

When plotting several points, it helps to:

  • Plot all points first
  • Label each point clearly
  • Identify patterns (collinear points, shapes)

Connecting Points:

Points can be connected to form geometric shapes:

  • Triangles (3 points)
  • Rectangles and squares (4 points)
  • Other polygons
Section Formula — Point Dividing a SegmentA(x₁,y₁)B(x₂,y₂)P(x,y)mnInternal Division (m:n):x = (mx₂ + nx₁)/(m+n) y = (my₂ + ny₁)/(m+n)Midpoint Formula (m = n = 1):M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )Example: A(2,3), B(8,9) divide in ratio 1:2x = (1×8 + 2×2)/(1+2) = 12/3 = 4 ; y = (1×9 + 2×3)/3 = 5 → P(4,5)
Example 1: What is the ordinate of (7, −2)?
The ordinate is the y-coordinate, −2.
Example 2: Are (2, 3) and (3, 2) the same point?
No — coordinates are ordered, so they are different points.
Quick recap
  • Abscissa = x (horizontal); ordinate = y (vertical).
  • Order matters: (x, y) ≠ (y, x) in general.
✓ Quick check
The base BC of a triangle lies along the x-axis with the origin as its midpoint. If B is (−3, 0), then C is:
If the origin is the midpoint and B is 3 units left at (−3, 0), then C is 3 units right at (3, 0).
A point P is 4 units from the x-axis and 3 units from the y-axis, in Quadrant III. Its coordinates are:
|y| = 4 and |x| = 3; in Quadrant III both are negative, so (−3, −4).

Distance Between Points (Pythagoras)

For points sharing a coordinate, the distance is just the difference of the other coordinate. For a general pair, drop a right triangle: the horizontal leg is the difference in x, the vertical leg the difference in y, and the distance is the hypotenuse by Pythagoras.

So the gap between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].

Example 1: Distance between (1, 2) and (4, 2)?
Same y, so it is just 4 − 1 = 3 units.
Example 2: Distance between (0, 0) and (3, 4)?
√(3² + 4²) = √25 = 5 units.
Quick recap
  • Same-row/column points: subtract the differing coordinate.
  • General distance = √[(Δx)² + (Δy)²] (Pythagoras).
✓ Quick check
Rohan sits at (3, 4). He is moved to a seat with the same abscissa but with the ordinate reduced by 5. His new position is:
Abscissa stays 3 and ordinate becomes 4 − 5 = −1, giving (3, −1).
Station Alpha is at (−4, 2) and Station Beta at (6, 2). A hub lies exactly halfway between them at:
Midpoint x = (−4 + 6)/2 = 1, y = 2, so the hub is (1, 2).
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