Endpoint Calculator

What Is an Endpoint in Geometry?

An endpoint is a point that marks the beginning or end of a line segment, ray, or vector. When working in coordinate geometry, a line segment $AB$ connects two endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$. If the location of one endpoint $A$ and the midpoint $M(x_m, y_m)$ are known, the missing endpoint $B$ can be determined algebraically by rearranging the standard midpoint formula.

The Endpoint Formula

The midpoint $M(x_m, y_m)$ divides a segment into two equal halves such that:

$$x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}$$

Multiplying both sides by $2$ and solving for the coordinates of the missing endpoint $(x_2, y_2)$ gives the Endpoint Formula:

$$x_2 = 2x_m - x_1$$

$$y_2 = 2y_m - y_1$$

3D Space Endpoint Formula

In three-dimensional Cartesian space with coordinates $(x, y, z)$, the $z$-coordinate follows the exact same relationship:

$$z_2 = 2z_m - z_1$$

Section Formula (General Division of a Line Segment)

If a point $P(x_p, y_p)$ divides the line segment $AB$ internally in a given ratio $m : n$ (so that $AP : PB = m : n$), the coordinates of $P$ are:

$$x_p = \frac{m x_2 + n x_1}{m + n}, \quad y_p = \frac{m y_2 + n y_1}{m + n}$$

When solving for the missing endpoint $B(x_2, y_2)$ from $A$ and $P$:

$$x_2 = \frac{(m + n)x_p - n x_1}{m}, \quad y_2 = \frac{(m + n)y_p - n y_1}{m}$$

Additional Properties of the Segment

Property	2D Formula	3D Formula
Segment Length $d(A, B)$	$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
Direction Vector $\vec{AB}$	$\langle x_2 - x_1, y_2 - y_1 \rangle$	$\langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle$
Slope $m$	$\frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2)$	N/A (Direction angles)
Line Equation	$y - y_1 = m(x - x_1)$	$\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$

Related Coordinate Geometry Tools

Distance Calculator: Find the distance between coordinates across multiple dimensions.
2D Distance Calculator: Calculate 2D Euclidean distance, Manhattan distance, and angle.
3D Distance Calculator: Compute distances between two 3D spatial points.
Slope Calculator: Calculate line slope, gradient, and angle of inclination.

Frequently Asked Questions

How do you find an endpoint given a start point and midpoint?

Multiply each coordinate of the midpoint by 2 and subtract the corresponding coordinate of the starting point: $x_2 = 2x_m - x_1$ and $y_2 = 2y_m - y_1$.

Can coordinates of endpoints be negative?

Yes, Cartesian coordinates can be negative, zero, or positive depending on which quadrant or octant the point lies in.

How do you find the endpoint in 3D coordinate space?

The formula extends directly to three dimensions: $x_2 = 2x_m - x_1$, $y_2 = 2y_m - y_1$, and $z_2 = 2z_m - z_1$.

What if the line segment is vertical?

If $x_1 = x_m$, then $x_2 = 2x_1 - x_1 = x_1$. The line is vertical, with an undefined slope, and its equation is simply $x = x_1$.

What is the difference between an endpoint and a midpoint?

An endpoint is located at the extreme boundary of a line segment, whereas the midpoint is the exact center point equidistant from both endpoints.

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