# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/geometric-transformation/reflection-over-line
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/geometric-transformation/reflection-over-line/en.mdx

Output docs content for large language models.

---

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

export const metadata = {
  title: "Reflection over a Line",
  description: "Master geometric reflection over any line. Understand mirror lines, perpendicular bisectors, and point transformation with visual examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/10/2025",
  subject: "Geometric Transformation",
};

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

1.  **If point <InlineMath math="P" /> lies on the mirror line <InlineMath math="m" />**:

    Its image is the point itself.

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

2.  **If point <InlineMath math="P" /> does not lie on the mirror line <InlineMath math="m" />**:

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

    <div className="flex flex-col gap-4">
      <BlockMath math="\text{Line segment } PP' \perp m" />
      <BlockMath math="\text{Distance}(P, m) = \text{Distance}(P', m)" />
    </div>

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

## 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:

<LineEquation
  title="Illustration of the Concept of Point Reflection over a Line"
  description={
    <>
      Point <InlineMath math="P(2,4)" /> is reflected over line{" "}
      <InlineMath math="m" /> 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.
