Find a missing endpoint given a midpoint and the other endpoint in 2D or 3D. Distance verification, calculation steps, presets for common problems, coordinate visualization, and formulas.
If you know a midpoint M and one endpoint A of a line segment, you can find the other endpoint B using the midpoint formula in reverse: B = 2M − A. In 2D, this means Bx = 2Mx − Ax and By = 2My − Ay. In 3D, the same logic extends with a z-coordinate. This is a common coordinate geometry problem in algebra, geometry, and physics courses. For example, if A = (1, 3) and M = (4, 5), then B = (2×4 − 1, 2×5 − 3) = (7, 7). This calculator handles both 2D and 3D cases, shows the step-by-step calculation, verifies the result by recomputing the midpoint and the distances from each endpoint to the midpoint (which must be equal), and plots the points on a coordinate grid. Presets include integer, fractional, and negative coordinate examples, plus 3D cases. Check the example with realistic values before reporting.
Finding a missing endpoint from a midpoint is a standard coordinate geometry problem that students solve frequently in algebra and geometry courses. This calculator provides instant answers with step-by-step reasoning (showing B = 2M − A applied to each coordinate), automatic midpoint verification, distance computation, and a coordinate grid visual — making it an ideal homework checker and teaching aid for 2D and 3D problems.
2D: B = (2Mx − Ax, 2My − Ay) 3D: B = (2Mx − Ax, 2My − Ay, 2Mz − Az) Verification: midpoint of A and B should equal M Distance = √((Bx−Ax)² + (By−Ay)²)
Result: Endpoint B = (7, 7)
Bx = 2(4) − 1 = 7, By = 2(5) − 3 = 7. Verification: midpoint of (1,3) and (7,7) = (4,5) ✓.
The midpoint formula says M = (A + B)/2. To find B when you know A and M, multiply both sides by 2: 2M = A + B, then subtract A: B = 2M − A. Applying this to each coordinate gives Bx = 2Mx − Ax, By = 2My − Ay, and (in 3D) Bz = 2Mz − Az. This derivation is worth understanding because the same "solve for the unknown" approach applies to weighted averages, centroids, and section formulas.
The midpoint is a special case of the section formula, which divides a segment in ratio m:n. The point P dividing AB in ratio m:n is P = (nA + mB)/(m+n). For the midpoint, m = n = 1, giving P = (A + B)/2. If you know P and A and the ratio, you can solve for B using the same algebraic approach. This extends the endpoint formula to any division ratio, useful in computer graphics for interpolation and in physics for center-of-mass calculations.
Endpoint problems appear in many geometric contexts: finding the other vertex of a parallelogram given the midpoint of a diagonal, reflecting a point across a line (the line is the perpendicular bisector), and computing symmetric points. In navigation, if you know your starting position and the midpoint of a route, you can calculate the destination using B = 2M − A with latitude and longitude coordinates (for small distances where curvature is negligible).
Use B = 2M − A. Each coordinate of the missing endpoint is twice the midpoint coordinate minus the known endpoint coordinate.
The midpoint formula says M = (A + B)/2. Solving for B gives B = 2M − A.
Yes — apply the same formula to each coordinate: Bx = 2Mx − Ax, By = 2My − Ay, Bz = 2Mz − Az. Use this as a practical reminder before finalizing the result.
Compute the midpoint of A and B: ((Ax+Bx)/2, (Ay+By)/2). It should equal your given midpoint M.
Use the distance formula: d = √((Bx−Ax)² + (By−Ay)²). Each endpoint is exactly d/2 from the midpoint.
Absolutely. The formula works for all real numbers.