Endpoint Calculator

The endpoint formula: B = 2M - A

When a segment has one end at A and its middle at M, the far end B is 2M - A, worked once per coordinate. Take the x numbers on their own, then the y numbers on their own. With M at (5, -3) and A at (8, 2), the x of B is 2 x 5 - 8 = 2 and the y of B is 2 x (-3) - 2 = -8, so B sits at (2, -8).

Order matters. Doubling the midpoint and subtracting the known end gives the missing end. Doubling the known end and subtracting the midpoint gives a point past A, twice as far from M as A is, which is a common slip on homework and on job sites alike.

The calculator above takes 2D or 3D points, accepts negative and decimal coordinates, and can run the other way, turning two endpoints into a midpoint. It also reports the segment length, the slope, the line equation and a check on a third point, so the arithmetic is easy to audit instead of taken on faith.

Where B = 2M - A comes from

The midpoint formula is an average: M = ((x1 + x2) / 2, (y1 + y2) / 2). Multiply both sides by 2 and you get 2M = (x1 + x2, y1 + y2). Subtract the known end from both sides and the unknown stands alone: x2 = 2Mx - x1 and y2 = 2My - y1. No new idea is needed, just one algebra step on a formula most people already know.

Read the result as a distance statement. The midpoint sits the same distance from each end, so the trip from A to M covers half the segment. Walk the same distance again in the same direction and you reach B. In vector form M - A = B - M, and rearranging gives B = 2M - A.

That is why the check works. Average A and B and you must land back on M, every time. The Purdue MA 15200 lesson notes list the midpoint and distance formulas on one page, and this Ashworth College geometry handout works an endpoint problem from a midpoint and one end in its second example.

Add the three squared gaps and take the square root for the length. In 2D drop the z column and the sum has two terms instead of three.

Worked example: midpoint (5, -3), endpoint (8, 2)

Work one coordinate at a time. x: 2 x 5 - 8 = 2. y: 2 x (-3) - 2 = -8. The other endpoint is (2, -8). Nothing in that pair of lines needs a diagram, which is the point of the formula.

Check it with the midpoint formula. The average of the x values is (8 + 2) / 2 = 5, and the average of the y values is (2 + -8) / 2 = -3. Both match the midpoint you started with, so the answer holds.

The segment is 11.6619 units long. The x gap is 8 - 2 = 6, the y gap is 2 - (-8) = 10, and the square root of 36 + 100 = 136 is 11.6619. Half of that is 5.831, the distance from M to either end. The slope is 10 / 6 = 1.6667, so the line reads y = 1.6667x - 11.3333 and it passes through all three points.

Negative and decimal coordinates

Sign errors show up once a coordinate goes below zero, and the fix is to slow down on the subtraction. Midpoint (-4.5, 6) with endpoint A at (2, -3): x is 2 x (-4.5) - 2 = -11 and y is 2 x 6 - (-3) = 15, so B is (-11, 15). The double minus in the y line is the step people drop, and dropping it turns 15 into 9.

Decimals behave the same way. Midpoint (2.5, -1.5) with A at (6, 4): x is 2 x 2.5 - 6 = -1 and y is 2 x (-1.5) - 4 = -7, so B is (-1, -7). Keeping the decimals instead of rounding early keeps the answer exact.

A quick test that catches most slips: B should be as far from M as A is, on the other side. From (2, -3) to (-4.5, 6) the steps are -6.5 in x and +9 in y. From (-4.5, 6) to (-11, 15) the steps are -6.5 and +9 again. Matching steps in both coordinates means the midpoint is where it should be.

Three dimensions: one more component

Add a z line and nothing else changes: z2 = 2Mz - z1. With M at (1, -2, 4) and A at (3, 0, -1), the answers are x = 2 - 3 = -1, y = -4 - 0 = -4 and z = 8 - (-1) = 9, so B is (-1, -4, 9).

Distance uses all three gaps: the square root of 16 + 16 + 100 = 132, which is 11.4891. Average each coordinate with its partner and you land back on (1, -2, 4), so the same check that works in 2D works here.

One thing 3D does not give you is a single slope. A line in space has a direction vector instead of a rise over run, so the calculator reports the direction rather than inventing a slope. Watch for a z value that quietly stays at 0: a 3D point typed into a 2D form flattens onto the plane and the answer looks right while being wrong.

Length, slope and the halfway check

Three numbers keep the answer honest. Length is the square root of (x2 - x1) squared plus (y2 - y1) squared. Half length is that divided by 2, and it should match the distance from either end to the midpoint. Slope is the y gap over the x gap, and a vertical line has none.

For the (8, 2) to (2, -8) case: x gap -6, y gap -10, length 11.6619, half length 5.831, slope 1.6667. Multiply the slope by the x gap and you get back the y gap: 1.6667 x -6 = -10.0. That single multiplication is a fast way to catch a flipped sign before it reaches a drawing.

A vertical segment breaks the slope step because the x gap is zero. Length and midpoint still work, and the line is written x = c instead of y = mx + b. A segment of zero length has no direction at all: both ends are the same point, and the calculator says so instead of dividing by zero.

The last row is a 3D case. Its length uses three gaps, so it is easy to compare against a 2D answer and see whether the z term was included.

Where this comes up: courts, boards, maps

Court and field layout. A high school basketball court is 94 feet by 50 feet, so with the origin in one corner the center circle sits at (47, 25). The baseline point that pairs with (0, 25) is (94, 25), from 2 x 47 - 0 and 2 x 25 - 25. Crews mark those points before they snap a chalk line, because one wrong end throws the whole layout off.

Woodwork. To center a 72 inch board, the midpoint is at 36. If a hole has to sit 14 inches from the center, the two marks are 36 + 14 = 50 and 36 - 14 = 22, each 14 inches from the middle and 22 inches from one end.

Spreadsheets. Put x values in one column and y values in the next, then write =2*Mx - Ax once and drag it down. Every row gets its missing end, and the sheet doubles as a record of the check.

Maps, with a caveat. Averaging latitude and longitude gives the middle of the straight line on a flat drawing, which is what a site plan wants. It is not the halfway point along a curved route, and it fails at the dateline: the average of 170 and -170 is 0, while the real middle of those two longitudes is 180. For ground distance on a globe, use the distance calculator instead of the flat formula.

Mistakes that flip the answer

Five mistakes cover most wrong answers. Using the midpoint as if it were an endpoint puts B halfway between A and M. Swapping x and y rotates the answer. Dropping a double minus shortens one coordinate. Doubling the endpoint instead of the midpoint puts B past A. And mixing 2D with 3D flattens a point onto the plane.

Another person's work is easy to test with a third point. Three points are collinear when the cross product of two direction vectors is zero. Take A(1, 2), B(5, 10) and C(3, 6): the vector AB is (4, 8) and AC is (2, 4), and 4 x 4 - 8 x 2 = 0, so C sits on the line. Move C to (4, 7) and the cross product becomes 4 x 5 - 8 x 3 = -4, so it does not.

The K20 Center lesson Meet Me in the Middle builds the midpoint idea from population estimates and ends with real layout tasks, which is close to how the formula shows up on the job.