Midpoint Formula
The midpoint formula gives the coordinates of the point that lies exactly halfway between two given points on a line segment in the Cartesian plane. It is derived by averaging the x-coordinates and the y-coordinates of the two endpoints separately. The midpoint is used extensively in geometry proofs, construction of perpendicular bisectors, and in applications such as GPS interpolation and computer graphics.
Key Formula
M = ((x1 + x2) / 2, (y1 + y2) / 2)
LaTeX: M = \left(\dfrac{x_1 + x_2}{2},\; \dfrac{y_1 + y_2}{2}\right)
| Symbol | Meaning | Unit |
|---|---|---|
| M | Midpoint coordinates | unitless (coordinate pair) |
| x₁, y₁ | Coordinates of the first endpoint | unitless |
| x₂, y₂ | Coordinates of the second endpoint | unitless |
Worked Example
Problem
Find the midpoint of the line segment joining A(−3, 7) and B(9, −1). Then verify it lies on the segment.
Solution
Step 1 — Apply formula: M = ((−3 + 9)/2, (7 + (−1))/2) = (6/2, 6/2) = (3, 3). Step 2 — Verify: Distance AM = √((3−(−3))² + (3−7)²) = √(36+16) = √52 ≈ 7.21. Step 3 — Distance BM = √((3−9)² + (3−(−1))²) = √(36+16) = √52 ≈ 7.21. Equal, so M is the midpoint. ✓
Answer
Midpoint M = (3, 3)
Midpoint Formula Applied to Various Segments
| Point A | Point B | Midpoint M | Segment Length AB |
|---|---|---|---|
| (0, 0) | (6, 8) | (3, 4) | 10 |
| (−3, 7) | (9, −1) | (3, 3) | √208 ≈ 14.42 |
| (2, 5) | (8, 5) | (5, 5) | 6 (horizontal) |
| (4, 1) | (4, 9) | (4, 5) | 8 (vertical) |
| (−4, −3) | (4, 3) | (0, 0) | 10 |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Distance Formula
The distance formula gives the straight-line (Euclidean) distance between two points in the Cartesian plane using their coordinates. It is derived directly from the Pythagorean theorem by treating the horizontal and vertical separations as legs of a right triangle. The formula extends naturally to three dimensions as d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) and is fundamental in analytic geometry, physics, and data science.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system in which points are represented by numerical coordinates. It provides a powerful bridge between algebra and geometry, allowing geometric shapes to be described by equations and algebraic operations to yield geometric insights. Key concepts include plotting points, finding distances, midpoints, slopes, and the equations of lines, circles, and conic sections on the Cartesian plane.
Ellipse (geometry)
An ellipse is a closed conic section defined as the set of all points in a plane for which the sum of the distances to two fixed points (called foci) is constant. It resembles a flattened circle and is characterised by its semi-major axis a (longest radius) and semi-minor axis b (shortest radius), with eccentricity e = c/a where c = √(a² − b²) gives the focal distance. Ellipses appear throughout nature and physics, most famously as the shapes of planetary orbits as described by Kepler's first law.
"Midpoint" is a compound of Old English "mid" (middle) + "point" (from Latin "punctum", a dot or position). Its algebraic form as a coordinate formula follows directly from Descartes' coordinate geometry framework established in 1637.