Coordinate Geometry: Distance, Midpoint and Slope

Choose a formula by its output

For A = (x1,y1) and B = (x2,y2), distance is d = √[(x2 − x1)² + (y2 − y1)²]. Midpoint is M = ((x1 + x2)/2,(y1 + y2)/2). Slope is m = (y2 − y1)/(x2 − x1), provided the denominator is nonzero. Distance returns a nonnegative length, midpoint returns an ordered pair, and slope returns a signed ratio. Keeping those output types beside the formulas prevents substituting an average where a difference is required.

Run one pair through all three formulas

Take A = (−3,2) and B = (5,8). The changes are 8 horizontally and 6 vertically. Distance is √(64 + 36) = 10 coordinate units. Midpoint is ((−3 + 5)/2,(2 + 8)/2) = (1,5). Slope is 6/8 = 3/4. These three answers describe the same segment in different ways. The midpoint is 5 units from either endpoint, and moving four units to the right along the line raises y by three units.

Build a line equation from the result

Point-slope form is y − y1 = m(x − x1). Using A and m = 3/4 gives y − 2 = (3/4)(x + 3). Expanding produces y = (3/4)x + 17/4. Substitute x = 5 to check that y = 8, recovering B. An intercept is a coordinate where a line meets an axis; it is not the segment’s midpoint. For a vertical line, write x equal to its constant horizontal coordinate instead of forcing an undefined slope into this formula.

Add only formulas justified by the geometry

For a circle with centre (h,k) and radius r, the equation is (x − h)² + (y − k)² = r². The distance formula explains why every point on that circle is r units from its centre. A polygon’s area needs more than two arbitrary points, so it cannot be inferred from the two-point calculator alone. Likewise, parallel and perpendicular line rules concern directions: equal finite slopes indicate parallel directions, while perpendicular nonvertical lines have slopes whose product is −1. Horizontal and vertical lines form the perpendicular exception where one of the slopes is undefined, so the finite-product rule cannot be used.

Keep coordinate pairs and units together

An ordered pair is written (x,y), with the horizontal coordinate first. Label the points before substituting numbers, and use the same point order in every difference. Negative coordinates are positions relative to the chosen origin; they do not make a geometric length negative. If coordinates represent measured lengths, put both axes in the same length unit before using the ordinary distance formula. A graph whose horizontal axis is time and vertical axis is temperature can have a meaningful rate of change, but the square root of their squared differences is not an ordinary physical distance.

Check the answer against the original points

Plotting a small sketch helps distinguish a plausible calculation from an input-order mistake. A segment’s midpoint must lie between its endpoints, and its distance must be at least the absolute horizontal change and at least the absolute vertical change. Reversing the endpoints leaves distance and midpoint unchanged. It also leaves a defined slope unchanged because both differences reverse sign. Keep exact fractions or square roots until the requested final rounding step; rounding each coordinate before subtracting can noticeably change a short segment between nearby points.

Use a plane-coordinate tool for plane-coordinate data

The linked calculator accepts two numeric coordinate pairs. It does not read coordinates from a photograph, identify a street route, or transform latitude and longitude into a local map projection. Record how the coordinates were obtained, the axis scale and the requested output before entering them. For a drawing measured in centimetres, the resulting distance is in centimetres; for a unitless exercise, report coordinate units. The input fields contain numbers only, so retain the unit labels in your working rather than appending them to a numeric field.

Related guides

Distance Formula Questions with Worked Answers

Practise coordinate distance with negative values, irrational answers and an unknown coordinate, then check each result with the distance formula.

Find the Midpoint of AB and Verify Both Half-Lengths

Find the midpoint of segment AB by averaging coordinates, handle negative endpoints correctly, and verify that the point splits the segment equally.

Midpoint Coordinates and Recovering a Missing Endpoint

Average endpoint coordinates to find a midpoint, rearrange the equations to recover a missing endpoint, and distinguish a midpoint from other ratios.