Geometry Lab reference
Distance & midpoint
Pythagoras meets the coordinate plane.
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
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).
What to watch for
Keep subtraction order consistent for both coordinate differences. Negative coordinates are valid; squared differences still contribute positively to distance.
Make the formula move.
Explore this idea with real inputs.