How to use the midpoint calculator
- Choose Find the midpoint or Find a missing endpoint, then 2D or 3D.
- To find a midpoint, enter Point 1 (x₁, y₁) and Point 2 (x₂, y₂). To find an endpoint, enter the Midpoint M (Mx, My) and the Known endpoint (x₁, y₁).
- Read the answer at the top. The steps underneath show each average or subtraction with your numbers.
How the midpoint formula works
The midpoint of a segment is the average of its endpoints, one coordinate at a time: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2), as given in OpenStax Intermediate Algebra, section 11.1. In 3D, add (z₁ + z₂) / 2. To work backward, the midpoint is the average, so the missing endpoint is x₂ = 2 × Mx - x₁ and y₂ = 2 × My - y₁. The calculator uses exact fractions, so averaging 0.1 and 0.2 gives exactly 0.15. A point that splits the segment in another ratio is a weighted average, and the midpoint is the case where both weights are equal.
Worked example with the starting numbers
The page opens with Point 1 = (2, 4) and Point 2 = (8, 10). Mx = (2 + 8) / 2 = 10 / 2 = 5 and My = (4 + 10) / 2 = 14 / 2 = 7, so the midpoint is M = (5, 7). Switch to Find a missing endpoint and it opens with the midpoint (5, 7) and the known endpoint (2, 4): x₂ = 2 × 5 - 2 = 8 and y₂ = 2 × 7 - 4 = 10, which returns (8, 10). In 3D the starting z values are 3 and 9, so Mz = (3 + 9) / 2 = 6.
Limits and edge cases
Negative coordinates work the same way: the midpoint of (-3, 5) and (4, -7) is (0.5, -1). If both points are the same, the midpoint is that point. A blank box is flagged instead of being treated as 0, and each number may have up to 40 digits. This tool finds the center of a straight segment only. To measure the segment, use the distance formula calculator; for its steepness, the slope calculator; to average more than two numbers, the average calculator.
Midpoint FAQ
How do I find an endpoint from the midpoint?
Double the midpoint and subtract the known endpoint, one coordinate at a time: x₂ = 2 × Mx - x₁ and y₂ = 2 × My - y₁. With a midpoint of (5, 7) and a known endpoint of (2, 4), the other endpoint is (8, 10).
Does the midpoint formula work with negative coordinates?
Yes. Add the coordinates with their signs and divide by 2. The midpoint of (-3, 5) and (4, -7) is ((-3 + 4) / 2, (5 + (-7)) / 2) = (0.5, -1). Put negative numbers in parentheses when you substitute them.
Is the midpoint the same as the average?
Yes. The midpoint is the average of the two x values and the average of the two y values, which is why it sits exactly halfway between the endpoints.
Does the midpoint formula work in 3D?
Yes. Average the z values too: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2). Choose 3D above the boxes to add the z fields. For (2, 4, 3) and (8, 10, 9) the midpoint is (5, 7, 6).
How do I find the perpendicular bisector of a segment?
It passes through the midpoint with a slope that is the negative reciprocal of the slope of the segment. For (2, 4) and (8, 10) the midpoint is (5, 7) and the segment slope is 1, so the bisector is y - 7 = -1(x - 5), or y = -x + 12.
Source: OpenStax, Intermediate Algebra 2e, Distance and Midpoint Formulas; Circles.