8
10
8.485281
4.242641
6
6
8
10
8.485281
4.242641
6
6
The Endpoint Calculator finds the missing endpoint of a line segment when you know the midpoint and the other endpoint. This is the reverse of the midpoint calculation and is a frequently encountered problem in coordinate geometry, standardized tests, and practical applications.
The derivation is straightforward. If \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\) is the midpoint, and we know \(M\) and one endpoint \((x_1, y_1)\), we can solve for the unknown endpoint: $$x_2 = 2M_x - x_1 \qquad y_2 = 2M_y - y_1$$
This formula comes from rearranging the midpoint equation. Since the midpoint is the average of the two endpoints, the unknown endpoint is found by "reflecting" the known endpoint through the midpoint. In other words, the midpoint is the center of symmetry: the unknown endpoint is exactly as far from the midpoint as the known endpoint, but in the opposite direction.
This calculation appears in many practical scenarios. In surveying, if you know a midpoint marker and one boundary stake, you can determine where the opposite stake should be placed. In computer graphics, reflecting a point through another is a core transformation. In physics, if two particles are connected by a rigid rod and you know the center of mass position and one particle's location, you can find the other.
The calculator also reports the total segment length and the half-distance from either endpoint to the midpoint, which serves as a verification check—both endpoints should be equidistant from the midpoint.
This inverse midpoint technique generalizes to higher dimensions by applying the same formula to each coordinate independently. It also extends to the section formula: if the known point divides the segment in a ratio other than 1:1, a modified formula applies. However, the midpoint (1:1 ratio) is by far the most common case in practice and on exams.
Understanding this relationship between midpoints and endpoints strengthens your grasp of coordinate geometry and provides a useful tool for solving problems where partial information about line segments is given.
The endpoint formula is derived by rearranging the midpoint formula:
Given the midpoint \(M(M_x, M_y)\) and one endpoint \((x_1, y_1)\), the unknown endpoint \((x_2, y_2)\) is:
$$x_2 = 2M_x - x_1$$
$$y_2 = 2M_y - y_1$$
Derivation: From the midpoint formula \(M_x = \frac{x_1 + x_2}{2}\), multiply both sides by 2: \(2M_x = x_1 + x_2\), then subtract \(x_1\): \(x_2 = 2M_x - x_1\). The same applies to the y-coordinate.
Geometrically, the unknown endpoint is the reflection of the known endpoint across the midpoint.
The Unknown Endpoint \((X_2, Y_2)\) completes the line segment. You can verify the result by computing the midpoint of the known and found endpoints—it should match the given midpoint. The Total Segment Length is the distance between the two endpoints, and the Distance to Midpoint should be exactly half of the total length.
Inputs
Results
X₂ = 2(5) - 2 = 8, Y₂ = 2(7) - 4 = 10. Verify: midpoint of (2,4) and (8,10) = (5,7). Correct!
Inputs
Results
X₂ = 2(0) - 3 = -3, Y₂ = 2(-1) - 5 = -7. The unknown endpoint has negative coordinates.
The midpoint calculator finds the center given two endpoints. The endpoint calculator does the reverse: it finds the missing endpoint given the midpoint and one endpoint.
Yes. Compute the midpoint of the known endpoint and the calculated endpoint. It should match the given midpoint exactly.
No. Whether you provide endpoint A or endpoint B as the known point, the formula correctly finds the other endpoint. The midpoint is symmetric.
If \(M = (x_1, y_1)\), then \(x_2 = 2x_1 - x_1 = x_1\) and \(y_2 = y_1\). This means both endpoints are the same as the midpoint—the segment has zero length.
Yes. Apply the same formula to each coordinate: \(x_2 = 2M_x - x_1\), \(y_2 = 2M_y - y_1\), \(z_2 = 2M_z - z_1\).
Exactly. Finding the unknown endpoint is geometrically identical to reflecting the known endpoint through the midpoint. This reflection interpretation is very useful in geometry and transformations.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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