# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/geometric-transformation/reflection-over-point
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/geometric-transformation/reflection-over-point/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 Point",
  description: "Master point reflection (180° rotation). Learn half-turn transformations using coordinate formulas with step-by-step examples and visualizations.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/10/2025",
  subject: "Geometric Transformation",
};

## Understanding Reflection over a Point

Reflection over a point, often called a half-turn rotation (<InlineMath math="180^\circ" />), is a geometric transformation where each point on an object is mapped to a new position such that the center of reflection becomes the midpoint between the original point and its image.

Suppose the center of reflection is <InlineMath math="P(a,b)" />. If a point <InlineMath math="Q(x,y)" /> is reflected over point <InlineMath math="P" />, its image <InlineMath math="Q'(x',y')" /> will lie on the line passing through <InlineMath math="Q" /> and <InlineMath math="P" />, with <InlineMath math="P" /> as the midpoint of the segment <InlineMath math="QQ'" />.

### Rule for Reflection over a Point

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

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

Alternatively, it can be written as:

<BlockMath math="Q'(-x + 2a, -y + 2b)" />

This means the x-coordinate of the image is twice the x-coordinate of the center minus the original x-coordinate, and the same applies to the y-coordinate.

## Reflecting a Point over Another Point

Determine the image of a half-turn <InlineMath math="\sigma_{(2,3)}" /> for the point <InlineMath math="(5, 4)" />.

This means we are reflecting point <InlineMath math="Q(5,4)" /> over the center point <InlineMath math="P(2,3)" />.

Here, <InlineMath math="x=5" />, <InlineMath math="y=4" />, <InlineMath math="a=2" />, and <InlineMath math="b=3" />.

Using the formula:

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

Thus, the image of point <InlineMath math="(5,4)" /> is <InlineMath math="(-1,2)" />.

<LineEquation
  title={
    <>
      Image of Point <InlineMath math="Q(5,4)" /> over Point{" "}
      <InlineMath math="P(2,3)" />
    </>
  }
  description={
    <>
      Visualization of reflecting point <InlineMath math="Q(5,4)" /> over the
      center point <InlineMath math="P(2,3)" /> resulting in{" "}
      <InlineMath math="Q'(-1,2)" />.
    </>
  }
  data={[
    {
      points: [{ x: 2, y: 3, z: 0 }],
      color: getColor("ROSE"),
      showPoints: true,
      labels: [{ text: "P(2,3) - Center", at: 0, offset: [1.5, -0.5, 0] }],
    },
    {
      points: [{ x: 5, y: 4, z: 0 }],
      color: getColor("SKY"),
      showPoints: true,
      labels: [{ text: "Q(5,4) - Original", at: 0, offset: [0.3, 0.5, 0] }],
    },
    {
      points: [{ x: -1, y: 2, z: 0 }],
      color: getColor("EMERALD"),
      showPoints: true,
      labels: [{ text: "Q'(-1,2) - Image", at: 0, offset: [-0.7, -0.5, 0] }],
    },
    {
      points: [
        { x: 5, y: 4, z: 0 },
        { x: -1, y: 2, z: 0 },
      ],
      color: getColor("PINK"),
    }, // Line connecting Q to Q'
  ]}
  showZAxis={false}
/>

## Reflecting a Line over a Point

Determine the image of a half-turn <InlineMath math="\sigma_{(1,3)}" /> for the line <InlineMath math="l" /> with the equation <InlineMath math="2x - y + 3 = 0" />.

Take an arbitrary point <InlineMath math="(x,y)" /> on line <InlineMath math="l" />. Its image, <InlineMath math="(x',y')" />, after reflection over point <InlineMath math="P(1,3)" /> is:

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

Substitute <InlineMath math="x = 2 - x'" /> and <InlineMath math="y = 6 - y'" /> into the equation of line <InlineMath math="l" />:

<div className="flex flex-col gap-4">
  <BlockMath math="2(2 - x') - (6 - y') + 3 = 0" />
  <BlockMath math="4 - 2x' - 6 + y' + 3 = 0" />
  <BlockMath math="-2x' + y' + 1 = 0" />
</div>

Replacing <InlineMath math="x'" /> and <InlineMath math="y'" /> back to <InlineMath math="x" /> and <InlineMath math="y" />, the equation of the image line <InlineMath math="l'" /> is:

<BlockMath math="-2x + y + 1 = 0" />

Alternatively, it can be written as <InlineMath math="2x - y - 1 = 0" />.

<LineEquation
  title={
    <>
      Image of Line <InlineMath math="2x-y+3=0" /> over Point{" "}
      <InlineMath math="P(1,3)" />
    </>
  }
  description={
    <>
      Original line <InlineMath math="2x-y+3=0" /> reflected over point{" "}
      <InlineMath math="P(1,3)" /> results in image line{" "}
      <InlineMath math="2x-y-1=0" />.
    </>
  }
  data={[
    {
      points: [{ x: 1, y: 3, z: 0 }],
      color: getColor("ROSE"),
      showPoints: true,
      labels: [{ text: "P(1,3) - Center", at: 0, offset: [0.5, -0.5, 0] }],
    },
    {
      // Original Line: 2x - y + 3 = 0  => y = 2x + 3
      points: Array.from({ length: 11 }, (_, i) => {
        const xVal = i - 5;
        return { x: xVal, y: 2 * xVal + 3, z: 0 };
      }),
      color: getColor("SKY"),
      labels: [{ text: "2x-y+3=0", at: 5, offset: [-2, 0.5, 0] }],
    },
    {
      // Image Line: 2x - y - 1 = 0 => y = 2x - 1
      points: Array.from({ length: 11 }, (_, i) => {
        const xVal = i - 5;
        return { x: xVal, y: 2 * xVal - 1, z: 0 };
      }),
      color: getColor("EMERALD"),
      labels: [{ text: "2x-y-1=0", at: 5, offset: [2, -0.5, 0] }],
    },
  ]}
  showZAxis={false}
  cameraPosition={[1, 2, 15]}
/>

## Exercises

1.  Determine the image of a half-turn <InlineMath math="\sigma_{(-2,3)}" /> for the point <InlineMath math="(3, -4)" />.
2.  Point <InlineMath math="K(-5, 1)" /> is reflected over the origin <InlineMath math="O(0,0)" />. Determine the coordinates of its image!
3.  Determine the image of a half-turn <InlineMath math="\sigma_{(1,-3)}" /> for the line <InlineMath math="l" /> with the equation <InlineMath math="2x + 5y + 6 = 0" />.

### Key Answers

1.  Center <InlineMath math="P(-2,3)" />, point <InlineMath math="Q(3,-4)" />. So <InlineMath math="a=-2, b=3, x=3, y=-4" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="x' = 2a - x = 2(-2) - 3 = -4 - 3 = -7" />
      <BlockMath math="y' = 2b - y = 2(3) - (-4) = 6 + 4 = 10" />
    </div>

    Thus, the image of point <InlineMath math="Q(3,-4)" /> is <InlineMath math="(-7,10)" />.

2.  Center <InlineMath math="O(0,0)" />, point <InlineMath math="K(-5,1)" />. So <InlineMath math="a=0, b=0, x=-5, y=1" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="x' = 2(0) - (-5) = 0 + 5 = 5" />
      <BlockMath math="y' = 2(0) - 1 = 0 - 1 = -1" />
    </div>

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

3.  Center <InlineMath math="P(1,-3)" />. Line <InlineMath math="2x + 5y + 6 = 0" />.

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

    Substitute into the line equation:

    <div className="flex flex-col gap-4">
      <BlockMath math="2(2 - x') + 5(-6 - y') + 6 = 0" />
      <BlockMath math="4 - 2x' - 30 - 5y' + 6 = 0" />
      <BlockMath math="-2x' - 5y' - 20 = 0" />
    </div>

    Image line equation: <InlineMath math="-2x - 5y - 20 = 0" /> or <InlineMath math="2x + 5y + 20 = 0" />.
