Midpoint Calculator
The midpoint calculator averages the two x coordinates and the two y coordinates. For (-2, 5) and (7, -1) the midpoint formula gives (5/2, 2), exactly halfway along the segment.
Midpoint
- Midpoint Halfway between the two points: the average of the x coordinates and the average of the y coordinates.
- (5/2, 2)
- x of the midpoint (x1 + x2) / 2 = (-2 + 7) / 2 = 5 / 2
- 5/2
- y of the midpoint (y1 + y2) / 2 = (5 + (-1)) / 2 = 4 / 2 = 2
- 2
- Midpoint as decimals Exact: half of a whole number always ends in .5 or .0, so these decimals terminate.
- (2.5, 2)
- Distance from each endpoint The same from both ends, which is what makes it the midpoint. Each half is half the length of the segment.
- 3√13/2 = 5.4083269
- Length of the segment By the distance formula: sqrt(9^2 + (-6)^2) = sqrt(117).
- 3√13 = 10.816654
- Slope of the segment
- -2/3
- Perpendicular bisector The line through the midpoint at a right angle to the segment, with slope -1/m = 3/2. Every point on it is the same distance from both endpoints.
- y = (3/2)x - 7/4
- Coordinates are read as the decimals you typed and the midpoint is computed in exact fractions, so 0.1 and 0.2 average to exactly 0.15.
How the Distance Formula Calculator works
The midpoint formula finds the point exactly halfway between two others. Average the x coordinates, average the y coordinates, and the pair of averages is the midpoint:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
For the default points, (-2, 5) and (7, -1), the x coordinates add to 5 and halve to 5/2, and the y coordinates add to 4 and halve to 2. The midpoint is (5/2, 2), which is (2.5, 2) as decimals. This calculator keeps the fraction when the inputs are whole numbers, because a textbook answer is usually written that way, and gives the decimal beside it. Half of a whole number always ends in .5 or .0, so the decimal is exact too.
Why averaging works: moving from the first point to the second changes x by x2 - x1. Halfway there, x has changed by half of that, so it is x1 + (x2 - x1)/2, which simplifies to (x1 + x2)/2. The same argument works for y, and for z in three dimensions, which is why the midpoint formula has the same shape in any number of dimensions.
The midpoint is the same distance from both endpoints, and each half is half the length of the segment. For the default segment the full length is the square root of 9^2 + 6^2 = 117, which is 3√13, about 10.816654, so each half is about 5.408327.
Two related questions come up next. The perpendicular bisector is the line through the midpoint at a right angle to the segment; every point on it is equally far from both endpoints, which is how a circle's center is found from three points on it. The calculator writes its equation. And if you know one endpoint and the midpoint, the other endpoint is twice the midpoint minus the known endpoint: B = 2M - A. Choose that mode to run the formula backward.
Distance Formula Calculator FAQ
What is the midpoint formula?
M = ((x1 + x2) / 2, (y1 + y2) / 2): the average of the x coordinates and the average of the y coordinates. For (-2, 5) and (7, -1) it gives (5/2, 2).
How do I find the midpoint of a line segment?
Add the two x coordinates and divide by 2, then add the two y coordinates and divide by 2. The two results are the coordinates of the midpoint. Negative coordinates work the same way: (-2 + 7) / 2 = 5/2.
How do I find an endpoint from the midpoint?
Double the midpoint and subtract the endpoint you know: B = 2M - A. If A is (1, 2) and the midpoint is (4, 6), the other endpoint is (2 x 4 - 1, 2 x 6 - 2) = (7, 10).
Should the midpoint be a fraction or a decimal?
Either is exact. Half of a whole number is always a whole number or ends in .5, so 5/2 and 2.5 are the same value. Textbooks usually keep the fraction.
What is the perpendicular bisector?
The line through the midpoint at a right angle to the segment. Its slope is the negative reciprocal of the segment's slope, and every point on it is the same distance from both endpoints.