# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/geometric-transformation/reflection-over-y-equals-h
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/geometric-transformation/reflection-over-y-equals-h/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 y = h",
  description: "Understand horizontal line reflections y = h with worked examples and visual guides. Apply the P'(x, 2h-y) formula to solve geometry problems.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/10/2025",
  subject: "Geometric Transformation",
};

## Understanding Reflection over the Line y = h

Reflection over the horizontal line <InlineMath math="y = h" /> is a geometric transformation that maps each point of an object to a new position. The line <InlineMath math="y = h" /> acts as a mirror.

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

### Rule for Reflection over the Line y = h

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

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

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

## Reflecting a Point over the Line y = h

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

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

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

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

Thus, the image of point <InlineMath math="P(3,2)" /> is <InlineMath math="P'(3,4)" />.

Now, let's visualize this example.

<LineEquation
  title={
    <>
      Image of Point <InlineMath math="P(3,2)" /> over Line{" "}
      <InlineMath math="y=3" />
    </>
  }
  description={
    <>
      Visualization of the reflection of point <InlineMath math="P(3,2)" /> over
      the line <InlineMath math="y=3" /> resulting in{" "}
      <InlineMath math="P'(3,4)" />.
    </>
  }
  data={[
    {
      points: [
        { x: -5, y: 3, z: 0 },
        { x: 5, y: 3, z: 0 },
      ],
      color: getColor("INDIGO"),
      labels: [{ text: "y=3", at: 1, offset: [0.5, 0.5, 0] }],
    }, // Line y=3
    {
      points: [{ x: 3, y: 2, z: 0 }],
      color: getColor("EMERALD"),
      showPoints: true,
      labels: [{ text: "P(3,2)", at: 0, offset: [0.3, -0.5, 0] }],
    },
    {
      points: [{ x: 3, y: 4, z: 0 }],
      color: getColor("EMERALD"),
      showPoints: true,
      labels: [{ text: "P'(3,4)", at: 0, offset: [0.2, 0.3, 0] }],
    },
    {
      points: [
        { x: 3, y: 2, z: 0 },
        { x: 3, y: 4, z: 0 },
      ],
      color: getColor("PINK"),
    }, // Vertical helper line
  ]}
  showZAxis={false}
/>

## Exercises

1.  Determine the image of point <InlineMath math="P(5,-4)" /> by reflection over the line <InlineMath math="y=-2" />.
2.  A point <InlineMath math="Q(-1, -3)" /> is reflected over the line <InlineMath math="y=0" /> (X-axis). Determine the coordinates of its image!
3.  The image of a point <InlineMath math="R(x,y)" /> after reflection over the line <InlineMath math="y=4" /> is <InlineMath math="R'(-2,5)" />. Determine the coordinates of point R!

### Key Answers

1.  Given <InlineMath math="P(5,-4)" /> and the mirror line <InlineMath math="y=-2" />. So <InlineMath math="x=5, y=-4, h=-2" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="x' = x = 5" />
      <BlockMath math="y' = 2h - y = 2(-2) - (-4) = -4 + 4 = 0" />
    </div>

    Thus, the image of point P is <InlineMath math="P'(5,0)" />.

2.  Given <InlineMath math="Q(-1, -3)" /> and the mirror line <InlineMath math="y=0" />. So <InlineMath math="x=-1, y=-3, h=0" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="x' = x = -1" />
      <BlockMath math="y' = 2h - y = 2(0) - (-3) = 0 + 3 = 3" />
    </div>

    Thus, the image of point Q is <InlineMath math="Q'(-1,3)" />.

3.  Given the image <InlineMath math="R'(-2,5)" /> and the mirror line <InlineMath math="y=4" />. So <InlineMath math="x'=-2, y'=5, h=4" />.

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

    From <InlineMath math="x' = x" />, then <InlineMath math="x = -2" />.

    From <InlineMath math="y' = 2h - y" />, then <InlineMath math="5 = 2(4) - y" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="5 = 8 - y" />
      <BlockMath math="y = 8 - 5" />
      <BlockMath math="y = 3" />
    </div>

    Thus, the coordinates of point R are <InlineMath math="R(-2,3)" />.
