Endpoint Calculator

What is an endpoint?

A line segment is the straight piece of a line between two points. Those two points are its endpoints; they mark where the segment starts and stops. On a coordinate plane each endpoint is written as a pair (x, y).

The midpoint is the point exactly halfway between the endpoints. It splits the segment into two equal parts, so it is the same distance from either end. If you know one endpoint and the midpoint, the other endpoint is fully determined: you walk from the known end to the midpoint, then keep going the same distance and in the same direction.

Endpoint formula

The midpoint M of a segment with endpoints (x₁, y₁) and (x₂, y₂) is found by averaging the coordinates:

M = ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 )

Turning this around gives the missing endpoint. If the known endpoint is (x₁, y₁) and the midpoint is (xₘ, yₘ), then the other endpoint is:

( x₂ , y₂ ) = ( 2 xₘ − x₁ , 2 yₘ − y₁ )Double the midpoint, then subtract the known endpoint, one coordinate at a time.

The x and y coordinates never interact, so you can treat them as two separate one-dimensional problems. The same idea extends to three dimensions: z₂ = 2 zₘ − z₁.

How to use the calculator

1. Enter the x and y coordinates of the point you already have in Starting point coordinates.
2. Enter the x and y coordinates of the middle of the segment in Midpoint coordinates.
3. Press Calculate endpoint. The result is shown as a pair, for example (9, 14).

Coordinates can be negative or have decimals. All four boxes must be filled in with plain numbers; exponent notation such as 1e2 is not accepted. Whole-number coordinates are shown as they are, values with one or two decimals are shown with two decimals (3.5 becomes 3.50), and longer decimals are rounded to four places.

Worked examples

Example 1. One end is (2, −3) and the midpoint is (5, 1). The other end is (2 × 5 − 2, 2 × 1 − (−3)) = (10 − 2, 2 + 3) = (8, 5). Check: the average of 2 and 8 is 5, and the average of −3 and 5 is 1. The segment's length is √(6² + 8²) = 10, so each half is 5 long.

Example 2: negative coordinates and decimals. One end is (−4, 7) and the midpoint is (0.5, 2.5). Then x = 2 × 0.5 − (−4) = 5 and y = 2 × 2.5 − 7 = −2, giving (5, −2).

Example 3: decimal results. One end is (1.5, 2.25) and the midpoint is (3, 4). The other end is (6 − 1.5, 8 − 2.25) = (4.5, 5.75), which the calculator prints as (4.50, 5.75).

Example 4: a circle's diameter. A circle has centre (1, 1), and one end of a diameter is (−3, 4). The centre is the midpoint of every diameter, so the opposite end is (2 − (−3), 2 − 4) = (5, −2).

Endpoints in geometry

The same calculation turns up whenever something is "balanced" about a point:

• Diameters. The centre of a circle bisects every diameter, so one end and the centre give the other end, as in Example 4.
• Point reflection. Reflecting a point through a centre P gives the point P′ for which P is the midpoint of the segment from the original point to P′. The calculator is a point-reflection tool.
• Parallelograms. The diagonals of a parallelogram bisect each other. Knowing one vertex and the intersection of the diagonals gives the opposite vertex.
• Graphics and CAD. Mirroring a shape about a pivot, or extending a line symmetrically, uses exactly this arithmetic.

To go the other way, from two endpoints to the middle, use the midpoint calculator.

Common mistakes

• Treating the midpoint as the other endpoint. The second row of the form is the middle of the segment, not its far end.
• Forgetting the subtraction. Doubling the midpoint alone gives the sum of the two endpoints, not the missing one.
• Mixing up x and y. Work each coordinate separately and keep the pairs in the same order.
• Sign errors with negatives. Subtracting a negative coordinate adds its size, as in 2 × 1 − (−3) = 5.