The Coordinate Plane
Two perpendicular number lines make up the coordinate plane: the horizontal x-axis and the vertical y-axis, crossing at the origin (0, 0). Any point on the plane gets an ordered pair (x, y), where x is the distance left or right of the origin and y is the distance up or down.
| Quadrant | x sign | y sign | Example |
|---|---|---|---|
| I (top-right) | + | + | (3, 5) |
| II (top-left) | − | + | (−2, 4) |
| III (bottom-left) | − | − | (−3, −1) |
| IV (bottom-right) | + | − | (4, −2) |
Distance Between Two Points
This formula is really the Pythagorean theorem applied to two arbitrary points, (x₁, y₁) and (x₂, y₂):
Distance from (1, 2) to (4, 6):
d = √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5
Midpoint of a Line Segment
Finding the point exactly halfway between two others just means averaging their coordinates separately, x with x and y with y:
Midpoint of (2, 3) and (8, 7):
= ((2+8)/2, (3+7)/2) = (5, 5)
Slope of a Line
Slope is a measure of steepness, calculated as rise over run. A line with positive slope climbs as you move left to right; one with negative slope drops.
Slope through (1, 2) and (4, 8):
m = (8 − 2) / (4 − 1) = 6 / 3 = 2
| Slope value | Meaning |
|---|---|
| Positive (e.g. 2) | Line rises left to right |
| Negative (e.g. −3) | Line falls left to right |
| Zero | Horizontal line |
| Undefined | Vertical line (x₂ = x₁) |
Equation of a Line
Slope-Intercept Form
This is the form you'll see most often: m gives the slope, and b marks the y-intercept, the point where the line crosses the y-axis:
Line with slope 3 and y-intercept −2: y = 3x − 2
Point-Slope Form
Reach for this form when you have a slope and a single point on the line, but not the y-intercept:
Slope = 2, point (3, 5):
y − 5 = 2(x − 3) → y = 2x − 1
Standard Form
A third way to write the same line, with both variables placed on one side of the equation:
Example: 3x − 2y = 6
Parallel and Perpendicular Lines
Two lines are parallel when they share the same slope, which is why they never intersect. Two lines are perpendicular when they meet at a right angle, and their slopes always multiply together to give −1.
Perpendicular: m₁ × m₂ = −1
If slope = 3, perpendicular slope = −1/3
Plotting It Automatically
SolverCalc's graphing calculator will plot any linear equation you give it. Type it in y = mx + b form and it draws the line instantly on a labeled grid, axes and all.
Key Takeaways
At its core, coordinate geometry converts drawing problems into arithmetic problems. Keep these four building blocks straight: distance comes from the Pythagorean theorem, midpoint comes from averaging, slope is rise over run, and a line is written as y = mx + b. Get comfortable with those four and most coordinate geometry questions you'll run into at school become routine.