Coordinate Geometry

What you'll learn

Describe the Cartesian plane, its axes, and the four quadrants
Plot points precisely given their coordinates
Read coordinates from a graph (abscissa and ordinate)
Find distances on axes and identify special positions
Answer questions about points lying on axes or at the origin

Key concepts

The Cartesian Plane

The Cartesian plane (coordinate plane) is formed by two perpendicular number lines intersecting at a point called the origin.

Key elements:

Element	Description
x-axis	Horizontal number line
y-axis	Vertical number line
Origin (O)	Point of intersection; coordinates (0, 0)
Coordinate	Ordered pair (x, y) describing a point's location

Named after: René Descartes (1596–1650), French mathematician and philosopher.

How to describe a point:

(x, y) — ordered pair; x comes first, y comes second.
x-coordinate = horizontal distance from the y-axis
y-coordinate = vertical distance from the x-axis

Axes and Quadrants

The two axes divide the plane into four quadrants.

          y
          |
  II      |      I
(−, +)    |    (+, +)
          |
──────────O──────────  x
          |
  III     |      IV
(−, −)    |    (+, −)
          |

Quadrant signs:

Quadrant	x-sign	y-sign	Example point
I (upper right)	+	+	(3, 5)
II (upper left)	−	+	(−4, 2)
III (lower left)	−	−	(−3, −6)
IV (lower right)	+	−	(7, −1)

Points on the axes:

Location	x-value	y-value	Example
On x-axis	Any value	0	(4, 0), (−2, 0)
On y-axis	0	Any value	(0, 3), (0, −5)
At origin	0	0	(0, 0)

Key rule: If a point lies on the x-axis, its y-coordinate is always 0. If it lies on the y-axis, its x-coordinate is always 0.

Plotting Points

Steps to plot point P(x, y):

Start at the origin O.
Move |x| units right (if x > 0) or left (if x < 0) along the x-axis.
From that position, move |y| units up (if y > 0) or down (if y < 0).
Mark and label the point.

Plotting example — plot A(3, 4), B(−2, 3), C(−4, −2), D(5, −3):

Point	Movement from O	Quadrant
A(3, 4)	3 right, 4 up	I
B(−2, 3)	2 left, 3 up	II
C(−4, −2)	4 left, 2 down	III
D(5, −3)	5 right, 3 down	IV

Special cases:

Point	Position
(0, 0)	Origin
(5, 0)	On positive x-axis
(−3, 0)	On negative x-axis
(0, 4)	On positive y-axis
(0, −7)	On negative y-axis

Distance on Axes

Distance between two points on the x-axis: Points (x₁, 0) and (x₂, 0): Distance = |x₂ − x₁|

Distance between (−3, 0) and (5, 0) = |5 − (−3)| = 8 units

Distance between two points on the y-axis: Points (0, y₁) and (0, y₂): Distance = |y₂ − y₁|

Distance between (0, −2) and (0, 7) = |7 − (−2)| = 9 units

Distance from the origin to a point on an axis:

(a, 0) from O: distance = |a|
(0, b) from O: distance = |b|

Abscissa and Ordinate

Abscissa = the x-coordinate of a point (horizontal position). Ordinate = the y-coordinate of a point (vertical position).

Term	Meaning	For point P(−3, 7)
Abscissa	x-coordinate	−3
Ordinate	y-coordinate	7

Reading coordinates from a graph — ordered pair (abscissa, ordinate):

Worked Example: A point is 4 units to the left of the y-axis and 2 units above the x-axis. Abscissa = −4, Ordinate = +2 → Point = (−4, 2) in Quadrant II.

Worked Example: A point Q has ordinate 0 and abscissa −5. y = 0 → Q lies on the x-axis. Q = (−5, 0)

Locating Points from Coordinates

From a word description to coordinates:

Description	Coordinates
3 units right of origin on x-axis	(3, 0)
5 units below x-axis on y-axis	(0, −5)
2 units left of y-axis, 4 units above x-axis	(−2, 4) — Quadrant II
Equal distance from both axes, in Quadrant III	(−a, −a) for some a > 0

Mirror / Reflection of points:

Original	Reflected in x-axis	Reflected in y-axis	Reflected in origin
(3, 4)	(3, −4)	(−3, 4)	(−3, −4)
(−2, 5)	(−2, −5)	(2, 5)	(2, −5)
(a, b)	(a, −b)	(−a, b)	(−a, −b)

Worked Example 1: In which quadrant does (−7, 0) lie? y = 0 → it lies on the negative x-axis (not in any quadrant).

Worked Example 2: The point (k, 3) lies in Quadrant II. What can you say about k? In Quadrant II, x-coordinate is negative → k < 0

Worked Example 3: A point P(a, b) satisfies a > 0 and b < 0. Where does P lie? Positive x, negative y → Quadrant IV

Worked Example 4: If a point (a, b) lies on the y-axis, what is the value of a? What does this tell you about the point? a = 0. The point has no horizontal displacement from the y-axis.

Collinear check (three points on a line):

Three points A, B, C are collinear if the area of the triangle they form = 0. Area = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Worked Example: Are A(1, 2), B(2, 4), C(3, 6) collinear? Area = ½|1(4−6) + 2(6−2) + 3(2−4)| = ½|−2 + 8 − 6| = ½|0| = 0 → Yes, collinear

Quick check

Plot and label the points P(−3, 5), Q(4, −2), R(0, −6), S(−5, 0). Name the quadrant or axis for each.
What is the abscissa of a point on the y-axis?
The distance between two points on the y-axis is 11 units. One point is (0, 4). Find the two possible positions of the second point.
In which quadrant does a point with negative abscissa and positive ordinate lie?
Are the points (2, 3), (4, 6), (6, 9) collinear? Show working.

Open the Practice tab for graded questions on Coordinate Geometry.