Geometry Lab formula guide
Distance & midpoint formulas
The distance between (x₁, y₁) and (x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²], which is the Pythagorean theorem on a grid. The midpoint averages the coordinates: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
| Use it to | Formula |
|---|---|
| Distance | d = √[(x₂ − x₁)² + (y₂ − y₁)²] |
| Midpoint | M = ((x₁ + x₂)/2, (y₁ + y₂)/2) |
Try it with your own numbers
Distance & midpoint formulas calculator
Pythagoras meets the coordinate plane. 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
The horizontal and vertical differences between two points form the legs of a right triangle. Their straight-line separation is its hypotenuse. To find the midpoint, average the two x-coordinates and the two y-coordinates separately.
Work through an example
1Work through an example
- For A(−3, −2) and B(4, 5), Δx = 7 and Δy = 7, giving distance √98 = 7√2 ≈ 9.899. The midpoint is ((−3 + 4)/2, (−2 + 5)/2) = (0.5, 1.5).
Common mistakes
- Keep subtraction order consistent for both coordinate differences. Negative coordinates are valid; squared differences still contribute positively to distance.
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