Formula and Method for the Endpoint Calculator
Given the midpoint of a line segment and one of its endpoints, you can always solve algebraically for the other endpoint. If M is the midpoint of a segment with endpoints A(x₁, y₁) and B(x₂, y₂), the midpoint formula states M = ((x₁+x₂)/2, (y₁+y₂)/2). Rearranging that equation to isolate the unknown endpoint gives the endpoint formula: x₂ = 2Mx − x₁ and y₂ = 2My − y₁. This calculator applies that formula directly, then reports the full segment length, the distance from the known endpoint to the midpoint, and the slope of the segment.
How the calculation works
Enter the coordinates of the endpoint you already know (x₁, y₁) and the coordinates of the midpoint (Mx, My). The calculator doubles each midpoint coordinate and subtracts the corresponding known-endpoint coordinate to get the missing endpoint: x₂ = 2Mx − x₁, y₂ = 2My − y₁. It then applies the distance formula, d = √((x₂−x₁)² + (y₂−y₁)²), to compute the full segment length between the two endpoints — this distance is always exactly twice the distance from the known endpoint to the midpoint, since the midpoint splits the segment into two equal halves. Finally, it reports the slope, (y₂−y₁)/(x₂−x₁), which is undefined when the segment is vertical.
Common mistakes
- Swapping the midpoint and endpoint: entering the midpoint's coordinates in the "known endpoint" fields (or vice versa) reverses the calculation and gives the wrong answer.
- Confusing this with the midpoint formula: if you already know both endpoints and want the point between them, use a midpoint calculator instead — this tool solves the reverse problem.
- Assuming the segment length equals the endpoint-to-midpoint distance: the midpoint is exactly halfway, so the full segment is twice as long as the distance from either endpoint to the midpoint.
Real-world applications
- Geometry and coordinate proofs use the endpoint formula to reconstruct a missing vertex when a midpoint and one vertex are known.
- Computer graphics and CAD tools use it to mirror or reflect a point through a center point.
- Surveying and mapping use it to locate an unknown boundary marker when the midpoint of a property line and one marker are already recorded.
- Physics problems use it to find a symmetric point along projectile or oscillation paths where the midpoint represents an equilibrium or turning point.