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URL: https://nakafa.com/en/subjects/mathematics/geometric-transformation/reflection-over-line
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/geometric-transformation/reflection-over-line/en.mdx

Learn geometric reflection over any line. Understand mirror lines, axes of symmetry, and point transformations through visual examples.

---

## What is Reflection over a Line?

Reflection over a line is a type of geometric transformation that moves every point on a plane to its mirror image position.

The line used as the reference for this reflection is called the **mirror line** or **axis of reflection**.

Imagine standing in front of a flat mirror. Your image in the mirror is the result of reflecting yourself over the surface of the mirror.

Mathematically, if we have a point $$P(x,y)$$ and a line $$m$$ as the axis of reflection, then the image point <InlineMath math="P'(x',y')" /> will have the following properties:

Visible text: Mathematically, if we have a point and a line as the axis of reflection, then the image point <InlineMath math="P'(x',y')" /> will have the following properties:

1.  **If point $$P$$ lies on the mirror line $$m$$**:

    Its image is the point itself.

    <BlockMath math="P' = P" />

2.  **If point $$P$$ does not lie on the mirror line $$m$$**:

    The line $$m$$ will be the perpendicular bisector of the line segment <InlineMath math="PP'" />. This means two important things:

    <MathContainer>
      <BlockMath math="\text{Line segment } PP' \perp m" />
      <BlockMath math="\text{Distance}(P, m) = \text{Distance}(P', m)" />
    </MathContainer>

    Consequently, the line $$m$$ intersects the line segment <InlineMath math="PP'" /> exactly at its midpoint. We call this intersection point $$M$$.

Visible text: 1. **If point lies on the mirror line **:

 Its image is the point itself.

 <BlockMath math="P' = P" />

2. **If point does not lie on the mirror line **:

 The line will be the perpendicular bisector of the line segment <InlineMath math="PP'" />. This means two important things:

 <MathContainer>
 <BlockMath math="\text{Line segment } PP' \perp m" />
 <BlockMath math="\text{Distance}(P, m) = \text{Distance}(P', m)" />
 </MathContainer>

 Consequently, the line intersects the line segment <InlineMath math="PP'" /> exactly at its midpoint. We call this intersection point .

## Visualization of Point Reflection over a Line

Let's try to visualize the concept of reflecting a point over a line using the following example:

Component: LineEquation
Props:
- title: Illustration of the Concept of Point Reflection over a Line
- description: Point $$P(2,4)$$ is reflected over line{" "}
$$m$$ to produce the image{" "}
<InlineMath math="P'(3.6, 0.8)" />.
  Visible text: Point is reflected over line{" "}
 to produce the image{" "}
<InlineMath math="P'(3.6, 0.8)" />.
- cameraPosition: [12, 8, 12]
- data: [
{
points: [
{ x: 0, y: 1, z: 0 },
{ x: 5, y: 3.5, z: 0 },
],
color: getColor("CYAN"),
lineWidth: 2,
labels: [
{
text: "m (mirror line)",
at: 1,
offset: [1.2, 0.5, 0],
},
],
},
{
points: [{ x: 2, y: 4, z: 0 }],
color: getColor("PURPLE"),
showPoints: true,
labels: [
{
text: "P(2,4) (original point)",
at: 0,
offset: [1.2, 0.5, 0],
},
],
},
{
points: [{ x: 3.6, y: 0.8, z: 0 }],
color: getColor("PURPLE"),
showPoints: true,
labels: [
{
text: "P'(3.6,0.8) (image point)",
at: 0,
offset: [1.7, -0.5, 0],
},
],
},
{
points: [{ x: 2.8, y: 2.4, z: 0 }],
color: getColor("PINK"),
showPoints: true,
labels: [
{
text: "M (midpoint)",
at: 0,
offset: [3.0, -0.3, 0],
},
],
},
{
points: [
{ x: 2, y: 4, z: 0 },
{ x: 3.6, y: 0.8, z: 0 },
],
color: getColor("PURPLE"),
labels: [{ text: "PP'", at: 0.5, offset: [-0.3, 0.3, 0] }],
},
]

This concept is fundamental to understanding how the coordinates of a point change after being reflected over various types of lines. The most important thing to remember is the geometric relationship between the original point, the image point, and the mirror line.