Geometry Lab formula guide
Slope & line equations
Slope is rise over run: m = (y₂ − y₁) / (x₂ − x₁). With the slope and one point, write the line as y = mx + b (slope-intercept form) or y − y₁ = m(x − x₁) (point-slope form).
m = (y₂ − y₁) / (x₂ − x₁)
| Use it to | Formula |
|---|---|
| Slope | m = (y₂ − y₁) / (x₂ − x₁) |
| Slope-intercept form | y = mx + b |
| Point-slope form | y − y₁ = m(x − x₁) |
| Parallel lines | m₁ = m₂ |
| Perpendicular lines | m₁ × m₂ = −1 (neither line vertical) |
Try it with your own numbers
Slope & line equations calculator
Describe a direction, then locate the line. Change a value and the diagram, result, and working update together.
Distance, midpoint & slopeLIVE EXPLORATION
Drag a labeled point · enter coordinates for precision
Your measurements
Drag either point on the grid, or enter its coordinates. Coincident points have no unique line.
Why it works
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
The idea
Slope is vertical change divided by horizontal change. Positive slope rises from left to right; negative slope falls. Once slope m is known, use a point to find b in y = mx + b, or write the point-slope form y − y₁ = m(x − x₁).
Work through an example
1Work through an example
- The line through (1, 2) and (4, 8) has slope (8 − 2)/(4 − 1) = 2. Substituting (1, 2) gives 2 = 2 × 1 + b, so b = 0 and the equation is y = 2x.
Common mistakes
- Vertical lines have undefined slope and equations such as x = 3. Horizontal lines have slope zero. Two identical points do not determine a unique line.
Make the formula move.
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