Math101 · Independent Ontario learning library
Math101learn.math101.caCoordinate Geometry
Coordinate geometry uses algebraic formulas for slope, distance, midpoint, and lines to analyze shapes on the plane.
Coordinates turn geometric relationships into quantities we can calculate and verify.
The coordinate plane
An ordered pair $(x,y)$ locates a point: move horizontally by $x$, then vertically by $y$. The axes divide the plane into four quadrants, with signs $(+,+)$, $(-,+)$, $(-,-)$, and $(+,-)$ counterclockwise from Quadrant I.
Order matters: $(2,-5)$ and $(-5,2)$ are different points.
Horizontal and vertical change
From $A(x_1,y_1)$ to $B(x_2,y_2)$, horizontal change is $x_2-x_1$ and vertical change is $y_2-y_1$.
These changes form a displacement vector and a right triangle whose hypotenuse joins the points.
Slope
For a nonvertical segment,
Slope describes steepness and direction. Horizontal lines have slope $0$; vertical lines have undefined slope because their run is zero.
Distance formula
The Pythagorean theorem gives
Distance is nonnegative and independent of endpoint order because both changes are squared.
Midpoint formula
The midpoint averages coordinates:
It lies halfway along the segment in both coordinate directions.
Worked example
One pair of points supplies direction, length, and centre information.
Equations of lines
Use point-slope form
when a point and slope are known. Convert to $y=mx+b$ or $Ax+By=C$ if useful.
A vertical line through $(c,y_0)$ is $x=c$.
Parallel and perpendicular lines
Distinct nonvertical parallel lines have equal slopes. Perpendicular nonvertical lines have negative-reciprocal slopes, so
Horizontal and vertical lines form a perpendicular special case.
Verifying shapes
Use slopes to show parallel or perpendicular sides, distances to show equal lengths, and midpoints to show diagonals bisect one another.
For example, a quadrilateral with four equal sides and one right angle is a square. State enough evidence for the classification; a picture alone is not proof.
Area from coordinates
A triangle's base and perpendicular height may be read from coordinates, or the shoelace formula can calculate polygon area from ordered vertices.
Sketch the vertex order to avoid self-crossing and include square units.
Transformations
Translations add fixed values to coordinates; reflections change signs or swap coordinates depending on the mirror line; rotations follow coordinate rules around the origin.
Coordinate transformations preserve or change length, angle, and orientation in predictable ways.
Common mistakes
Swapping $x$ and $y$. Ordered pairs always list horizontal first.
Reversing subtraction in only one slope component. Keep endpoint order consistent.
Using negative distance. Distance is a magnitude.
Calling equal slopes perpendicular. Equal slopes are parallel.
Classifying a shape from appearance. Use calculated evidence.
Quick self-check
- Are points plotted with correct order and signs?
- Do slope differences use consistent endpoint order?
- Are distance and midpoint formulas applied coordinate by coordinate?
- Do line equations contain the required point?
- Are parallel/perpendicular claims supported by slopes?
- Does a shape proof use enough lengths, slopes, or midpoints?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the distance between A = (−2, 3) and B = (4, −5).
- d = √(6² + (−8)²)
- d = √100 = 10.
End of lesson
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