# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/geometric-transformation/reflection-over-x-equals-k
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/geometric-transformation/reflection-over-x-equals-k/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 Line x = k",
  description: "Master reflection over vertical lines x = k with step-by-step examples. Learn the transformation rule P'(2k-x, y) and solve practice problems.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/10/2025",
  subject: "Geometric Transformation",
};

## Understanding Reflection over the Line x = k

Reflection over the vertical line <InlineMath math="x = k" /> is a geometric transformation where each point of an object is mapped to a new position. The line <InlineMath math="x = k" /> acts as a mirror.

The horizontal distance from the original point to the mirror line is equal to the horizontal distance from the image point to the mirror line. The y-coordinate of the point does not change.

### Rule for Reflection over the Line x = k

If a point <InlineMath math="P(x, y)" /> is reflected over the line <InlineMath math="x = k" />, its image's coordinates, <InlineMath math="P'(x', y')" />, are determined by the rule:

<div className="flex flex-col gap-4">
  <BlockMath math="x' = 2k - x" />
  <BlockMath math="y' = y" />
</div>

Thus, the image of point <InlineMath math="P(x, y)" /> is <InlineMath math="P'(2k - x, y)" />. Note that the y-coordinate remains the same, while the x-coordinate changes based on its distance from the line <InlineMath math="x=k" />.

## Reflecting a Point over the Line x = k

Determine the image of point <InlineMath math="P(3,2)" /> by reflection over the line <InlineMath math="x=3" />.

In this case, <InlineMath math="x=3" />, <InlineMath math="y=2" />, and <InlineMath math="k=3" />.

Using the rule <InlineMath math="P'(2k - x, y)" />:

<div className="flex flex-col gap-4">
  <BlockMath math="x' = 2(3) - 3 = 6 - 3 = 3" />
  <BlockMath math="y' = 2" />
</div>

Thus, the image of point <InlineMath math="P(3,2)" /> is <InlineMath math="P'(3,2)" />. The point lies on the mirror line, so its image is the point itself.

Now, let's try another example. Determine the image of point <InlineMath math="A(1,4)" /> if it is reflected over the line <InlineMath math="x=3" />.

Here, <InlineMath math="x=1" />, <InlineMath math="y=4" />, and <InlineMath math="k=3" />.

<div className="flex flex-col gap-4">
  <BlockMath math="x' = 2(3) - 1 = 6 - 1 = 5" />
  <BlockMath math="y' = 4" />
</div>

Thus, the image of point <InlineMath math="A(1,4)" /> is <InlineMath math="A'(5,4)" />.

<LineEquation
  title={
    <>
      Image of a Point over the Line <InlineMath math="x=k" />
    </>
  }
  description={
    <>
      Visualization of the reflection of point <InlineMath math="A(1,4)" /> over
      the line <InlineMath math="x=3" /> resulting in{" "}
      <InlineMath math="A'(5,4)" />.
    </>
  }
  data={[
    {
      points: [
        { x: 3, y: -5, z: 0 },
        { x: 3, y: 5, z: 0 },
      ],
      color: getColor("INDIGO"),
      labels: [{ text: "x=3", at: 1, offset: [0.5, 0.5, 0] }],
    }, // Line x=3
    {
      points: [{ x: 1, y: 4, z: 0 }],
      color: getColor("EMERALD"),
      showPoints: true,
      labels: [{ text: "A(1,4)", at: 0, offset: [-0.5, 0.3, 0] }],
    },
    {
      points: [{ x: 5, y: 4, z: 0 }],
      color: getColor("EMERALD"),
      showPoints: true,
      labels: [{ text: "A'(5,4)", at: 0, offset: [0.2, 0.3, 0] }],
    },
    {
      points: [
        { x: 1, y: 4, z: 0 },
        { x: 5, y: 4, z: 0 },
      ],
      color: getColor("PINK"),
    }, // Horizontal helper line
  ]}
  showZAxis={false}
/>

## Exercises

1.  Determine the image of point <InlineMath math="P(3,2)" /> by reflection over the line <InlineMath math="x=5" />.
2.  A point <InlineMath math="B(-2, 5)" /> is reflected over the line <InlineMath math="x=1" />. Determine the coordinates of its image!
3.  The image of a point <InlineMath math="C(x,y)" /> after reflection over the line <InlineMath math="x=-2" /> is <InlineMath math="C'(0,3)" />. Determine the coordinates of point C!

### Key Answers

1.  Given <InlineMath math="P(3,2)" /> and the mirror line <InlineMath math="x=5" />. So <InlineMath math="x=3, y=2, k=5" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="x' = 2k - x = 2(5) - 3 = 10 - 3 = 7" />
      <BlockMath math="y' = y = 2" />
    </div>

    Thus, the image of point P is <InlineMath math="P'(7,2)" />.

2.  Given <InlineMath math="B(-2, 5)" /> and the mirror line <InlineMath math="x=1" />. So <InlineMath math="x=-2, y=5, k=1" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="x' = 2k - x = 2(1) - (-2) = 2 + 2 = 4" />
      <BlockMath math="y' = y = 5" />
    </div>

    Thus, the image of point B is <InlineMath math="B'(4,5)" />.

3.  Given the image <InlineMath math="C'(0,3)" /> and the mirror line <InlineMath math="x=-2" />. So <InlineMath math="x'=0, y'=3, k=-2" />.

    We know <InlineMath math="x' = 2k - x" /> and <InlineMath math="y' = y" />.

    From <InlineMath math="y' = y" />, then <InlineMath math="y = 3" />.

    From <InlineMath math="x' = 2k - x" />, then <InlineMath math="0 = 2(-2) - x" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="0 = -4 - x" />
      <BlockMath math="x = -4" />
    </div>

    Thus, the coordinates of point C are <InlineMath math="C(-4,3)" />.
