Endpoint Calculator: Find the Missing Endpoint from the Midpoint or a Ratio
Find the other endpoint of a segment from one endpoint and the midpoint, or from a point that divides it in a ratio m:n, in 2D or 3D, with steps, lengths and line equation.
How it works
From the midpoint: x₂ = 2xₘ − x₁, y₂ = 2yₘ − y₁ (z₂ = 2zₘ − z₁) · from a point P with AP : PB = m : n: B = A + (P − A) × (m + n) ÷ m · section formula: P = (m × B + n × A) ÷ (m + n) · length AB = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
The midpoint is the average of the two endpoints, xₘ = (x₁ + x₂) ÷ 2. Solve that for x₂ and you get x₂ = 2xₘ − x₁: double the midpoint and subtract the endpoint you know, separately for each coordinate. Geometrically, you step from A to the midpoint and then take the same step again. The same idea works for any point on the segment. If P divides AB so that AP : PB = m : n, then P lies m ÷ (m + n) of the way from A, so B = A + (P − A) × (m + n) ÷ m; for a midpoint, m = n and this becomes 2M − A. The calculator works in 2D or 3D, shows each coordinate with its working, the segment length, and in 2D the slope and the equation of the line through A and B, plus a check that recomputes the midpoint. The third mode does the forward job (the section formula) when you know both endpoints and want the point at a given ratio. The midpoint calculator goes the other way from the first mode: it takes two endpoints and returns the midpoint, and the section-formula calculator covers only the forward ratio problem; this page is built around the missing endpoint.
Worked example
One endpoint is A(−2, 5) and the midpoint is M(3, 1). x₂ = 2 × 3 − (−2) = 8 and y₂ = 2 × 1 − 5 = −3, so B = (8, −3). Check: ((−2 + 8) ÷ 2, (5 + (−3)) ÷ 2) = (3, 1). The segment AB is √(10² + (−8)²) = √164 = 12.8062 long, AM is 6.4031, the slope is −8 ÷ 10 = −0.8, and the line is y = −0.8x + 3.4.
Sources: Omni Calculator: Endpoint calculator (x₂ = 2x − x₁, (1, 3) and (3, 5) example) · Omni Calculator: Ratios of directed line segments (section formula, A(1, 2), B(4, 6), 2 : 3) · Best-Calculators: Endpoint calculator (half and full segment length)
Frequently asked questions
How do you find an endpoint from the midpoint?
Double each midpoint coordinate and subtract the matching coordinate of the known endpoint: x₂ = 2xₘ − x₁ and y₂ = 2yₘ − y₁. With A(1, 3) and M(3, 5): x₂ = 6 − 1 = 5, y₂ = 10 − 3 = 7, so B = (5, 7).
What is the endpoint formula?
It is the midpoint formula solved for the missing endpoint: (x₂, y₂) = (2xₘ − x₁, 2yₘ − y₁). In 3D add z₂ = 2zₘ − z₁.
How do I find the endpoint if the point is not the midpoint?
If P divides AB in the ratio AP : PB = m : n, then B = A + (P − A) × (m + n) ÷ m for each coordinate. For A(1, 2), P(2.2, 3.6) and 2 : 3, B = (1 + 1.2 × 2.5, 2 + 1.6 × 2.5) = (4, 6).
How do you find a point that divides a segment in a given ratio?
Use the section formula P = (m × B + n × A) ÷ (m + n). For A(1, 2), B(4, 6) and 2 : 3, P = ((8 + 3) ÷ 5, (12 + 6) ÷ 5) = (2.2, 3.6).
Can the endpoint have negative coordinates?
Yes. If the midpoint is closer to the origin than the known endpoint, the other endpoint often lands on the far side, for example A(5, 2) and M(1, 1) give B(−3, 0).
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.