Reflect a point across a line and find its new coordinates. Enter the line coefficients and the original point to calculate the reflected point quickly.
📐 Line Reflection Calculator
Reflected point: (4, -3)
Line: x – y = 0
What Is Reflection Across a Line?
Reflection is a geometric transformation that produces a mirror image of a point or shape across a line.
If a point is reflected across a line, the original and reflected points are the same perpendicular distance from the line.
Line Format
This calculator represents the line as:
ax + by + c = 0
You enter:
- a — x coefficient
- b — y coefficient
- c — constant
- Px — x-coordinate of the point
- Py — y-coordinate of the point
Reflection Formula
For point (x₀, y₀) and line:
ax + by + c = 0
First calculate:
d = (ax₀ + by₀ + c) / (a² + b²)
The reflected point is:
x’ = x₀ − 2ad
y’ = y₀ − 2bd
Example
Suppose the line is:
x − y = 0
and the point is:
(2, 3)
The reflected point is:
(3, 2)
This makes sense because reflecting across y = x swaps the x and y coordinates.
Where Is Reflection Used?
Reflections are common in:
- Coordinate geometry
- Algebra
- Transformations
- Graphing
- Geometry homework
- Computer graphics
FAQs
Can the line be horizontal or vertical?
Yes, provided it is entered correctly in the required coefficient format.
What happens if a = b = 0?
That does not describe a valid line and should not be used.