Endpoint Calculator

1. Choose 2D or 3D. Most algebra problems are 2D. Pick 3D when the points have a z coordinate.
2. Type the known endpoint A and the midpoint M. Negative numbers and decimals are fine. Make sure you do not swap the two points.
3. Read endpoint B, then the check. The tool shows the arithmetic for each coordinate, a plot in 2D, and the midpoint of A and B, which must equal M.

An endpoint is one of the two points that bound a line segment. A segment AB starts at A and stops at B, and its midpoint M sits exactly halfway between them.

An endpoint problem runs the midpoint formula backwards. The midpoint calculator takes two endpoints and returns M. This tool takes one endpoint and M and returns the other endpoint. Both rely on the same fact: the midpoint coordinates are the averages of the endpoint coordinates.

OpenStax College Algebra 2e notes that the two pieces on either side of a midpoint are congruent, meaning equal in length. So AM = MB, and the full segment AB is twice AM.

An endpoint is not the same as the end of a ray or a line. A ray has one endpoint and a line has none; only a segment has two.

Start from the midpoint formula and solve for the unknown endpoint.

1. Write the x part of the midpoint formula: xM = (x1 + x2) / 2.
2. Multiply both sides by 2: 2 xM = x1 + x2.
3. Subtract the known x1: x2 = 2 xM - x1.
4. Repeat for y, and for z in 3D.
5. Check by putting both endpoints back into the midpoint formula.

Another way to see it: the step from A to M, (xM - x1, yM - y1), is half the trip. Take the same step again from M and you land on B.

One endpoint of a segment is A(2, 3) and its midpoint is M(5, -1). Find the other endpoint B.

1. x: 2 x 5 - 2 = 10 - 2 = 8.
2. y: 2 x (-1) - 3 = -2 - 3 = -5.
3. So B = (8, -5).
4. Check: ((2 + 8) / 2, (3 + (-5)) / 2) = (5, -1), which is M.

The step from A to M is (3, -4), so AM = sqrt(3² + 4²) = 5 and AB = 10. The slope of AB is -4 / 3 = -1.333333, and the line is y = -1.333333x + 5.666667.

A 3D Example

With A(1, 2, 3) and M(4, 0, 5), B = (2 x 4 - 1, 2 x 0 - 2, 2 x 5 - 3) = (7, -2, 7). The step is (3, -2, 2), so AM = sqrt(17) = 4.123106 and AB = 8.246211.

Which point is the midpoint, the signs of the coordinates and the number of dimensions all change the answer.

Which Point Is the Midpoint

Swapping A and M gives a different point. With M(2, 3) and A(5, -1), B would be (-1, 7), not (8, -5). Label the midpoint before you start.

Negative Coordinates

Doubling a negative midpoint coordinate and then subtracting a negative endpoint coordinate is where sign errors creep in. The method line puts brackets around every negative value to keep the signs visible.

2D vs 3D

The rule is the same in each coordinate, so 3D just adds a third line of arithmetic. The distance also gains a z term, as OpenStax Calculus Volume 3 shows for points in space.

Decimals and Fractions

Coordinates such as 2.5 or 1/3 work the same way. Type fractions as decimals; results show up to 6 decimal places.

Three related formulas answer three different questions about a segment.

OpenStax Intermediate Algebra 2e describes the midpoint as the average of the x coordinates and the average of the y coordinates. The endpoint rule simply undoes that average.

Geometry and Algebra Homework

"The midpoint of AB is (5, -1) and A is (2, 3); find B" is a standard textbook exercise. The worked steps match the layout teachers expect.

Circles and Diameters

The center of a circle is the midpoint of any diameter. With center (3, -2) and one end of a diameter at (7, 1), the other end is (-1, -5), and the radius is 5.

Reflections Through a Point

Reflecting a point A through a center point M gives exactly the endpoint B. Graphic designers and game developers use this for point symmetry.

Checking a Perpendicular Bisector

The perpendicular bisector of AB passes through M at a right angle to AB. A zero dot product between the AB direction and the bisector direction confirms the right angle; the dot product calculator does that step.

1. Averaging Instead of Reversing

Computing (2 + 5) / 2 = 3.5 finds the midpoint of A and M, a quarter of the way along AB. The endpoint needs 2M - A.

2. Subtracting in the Wrong Order

A - 2M gives the negative of the right answer in each coordinate. For the example it would give (-8, 5) instead of (8, -5).

3. Dropping a Negative Sign

2 x (-1) - 3 is -5, not -1. Write negative coordinates in brackets before multiplying.

4. Mixing Up X and Y

Keep each coordinate in its own line. The x of B only uses x values, and the y of B only uses y values.

5. Skipping the Check

Averaging A and B takes a few seconds and catches every one of the errors above. The tool shows this check on every result.

6. Forgetting the Third Coordinate

In 3D problems, a missing z coordinate leaves the point undefined. Switch to 3D and fill in both z values. To write the full line through A and B, the linear equation solver handles the algebra.

The endpoint is exact for the numbers you type; results are shown to 6 decimal places.

Double each midpoint coordinate and subtract the matching coordinate of the known endpoint: x2 = 2xM - x1 and y2 = 2yM - y1.

Change the sign of every coordinate. With M(0, 0) and A(4, -7), B is (-4, 7).

Yes. B = (8, -5) in the main example. Any coordinate can be negative, zero or positive.

Yes. Apply it to x, y and z separately. A(1, 2, 3) with M(4, 0, 5) gives B(7, -2, 7).

Average the known endpoint and your answer. If the result is the given midpoint, the answer is right.

Then the other endpoint is that same point too, and the segment has zero length.

It is twice the distance from the known endpoint to the midpoint. For A(2, 3) and M(5, -1), AM = 5, so AB = 10.

Use the center as the midpoint. Center (3, -2) with one end at (7, 1) gives the other end at (-1, -5).

Yes. The midpoint must go in the midpoint fields. Swapping them reflects the wrong point and gives a different answer.

No. The calculation runs in your browser; Save keeps a copy only in your own browser storage.

• The midpoint formula and the distance formula (OpenStax College Algebra 2e, section 2.1).
• Midpoint as the average of the x and y coordinates (OpenStax Intermediate Algebra 2e, section 11.1).
• Distance and midpoints for points in three dimensions (OpenStax Calculus Volume 3, section 2.2).