Midpoint Calculator

Find the midpoint of a line segment, work backward to find a missing endpoint, or find the point that divides a segment in any ratio m : n. Enter the coordinates and read the answer with steps. Example: the midpoint of (2, 3) and (8, 11) is (5, 7). If the midpoint is (5, 7) and one endpoint is (2, 3), the other endpoint is (8, 11).

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Distance & Midpoint Calculator

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Type the problems from this lesson and check each step.

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What do you want to find?

x₁ (first endpoint)

y₁ (first endpoint)

x₂ (second endpoint)

y₂ (second endpoint)

Midpoint x (for missing endpoint)

Midpoint y (for missing endpoint)

Ratio m

Ratio n

How to use the Midpoint Calculator

Choose what to find: the midpoint, a missing endpoint, or a point that divides the segment in a ratio.
For the midpoint, enter both endpoints. For a missing endpoint, enter one endpoint and the midpoint. For a ratio point, enter both endpoints and the ratio m : n.
Press Calculate to read the result and see the steps.

Formula and method

Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Missing endpoint: x₂ = 2x₀ − x₁ and y₂ = 2y₀ − y₁, where (x₀, y₀) is the midpoint.

Section formula (ratio m : n from the first point): ((n·x₁ + m·x₂) / (m + n), (n·y₁ + m·y₂) / (m + n)).

Example: endpoints (2, 3) and (8, 11) give ((2 + 8) / 2, (3 + 11) / 2) = (5, 7).

What is a midpoint?

The midpoint splits a segment into two equal parts. It is the average of the endpoints, taken separately for x and for y. The midpoint is also the centre of the circle that has the segment as a diameter.

Finding a missing endpoint

If M is the midpoint of AB and you know M and A, then B = 2M − A. Doubling the midpoint and subtracting the known endpoint reverses the averaging. Geometry problems often hide this as "M(5, 7) is the midpoint of AB and A is (2, 3)".

Dividing a segment in a ratio

For m : n = 1 : 1 the section formula gives the midpoint. For 1 : 2 the point is one third of the way from the first endpoint; for 3 : 1 it is three quarters of the way. This is useful in coordinate geometry for centroids, trisection points and weighted averages.

Ratio m : n	Fraction from the first point
1 : 1	1/2
1 : 2	1/3
2 : 1	2/3
1 : 3	1/4
3 : 1	3/4

Worked example

Find the point that is 2/5 of the way from A(1, 2) to B(11, 7). Use m : n = 2 : 3. x = (3·1 + 2·11) / 5 = 5; y = (3·2 + 2·7) / 5 = 4. The point is (5, 4).

Perpendicular bisector

The perpendicular bisector passes through the midpoint at a right angle to the segment. Its slope is the negative reciprocal of the segment’s slope. The calculator shows that slope in the midpoint mode.

Midpoint in 3D and on a number line

On a number line, the midpoint of a and b is (a + b) / 2. In 3D, average the third coordinate too. Use the distance calculator in 3D mode for the length of a space segment.

Common mistakes

Adding the coordinates without halving, mixing x with y, and dropping a negative sign. The midpoint always lies between the endpoints in each coordinate; if it does not, recheck the arithmetic.

Accuracy and limits

All results are exact arithmetic, shown to four decimals.

Frequently asked questions

How do you find the midpoint?

Average the x-coordinates and average the y-coordinates.

How do I find a missing endpoint?

Double the midpoint coordinate and subtract the known endpoint coordinate.

What is the midpoint of (2, 3) and (8, 11)?

(5, 7).

What is the section formula?

It finds the point dividing a segment in the ratio m : n: ((n·x₁ + m·x₂)/(m + n), (n·y₁ + m·y₂)/(m + n)).

Is the midpoint the same as the centroid?

For a segment, yes. For a triangle, the centroid is the average of the three vertices, which is not the midpoint of a side.

Sources

Midpoint (Wikipedia)

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Effortless Math Team

3 hours ago