# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/geometric-transformation/dilation
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/geometric-transformation/dilation/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: "Dilation",
  description: "Learn geometric dilation with scale factors and center points. Master enlargement & reduction formulas with visual examples and practice problems.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/10/2025",
  subject: "Geometric Transformation",
};

## Understanding Dilation

Dilation is a geometric transformation that changes the size of an object (enlarging or shrinking) without changing its shape. Each point on the object is mapped to a new position based on a center of dilation and a scale factor.

### Formal Definition of Dilation

Given a point <InlineMath math="C" /> as the center of dilation and a scale factor <InlineMath math="k \neq 0" />. The dilation of a point <InlineMath math="A" /> with respect to center <InlineMath math="C" /> by a factor <InlineMath math="k" />, denoted as <InlineMath math="D_{(C,k)}" />, is a transformation that maps <InlineMath math="A" /> to <InlineMath math="A' = D_{(C,k)}(A)" /> such that <InlineMath math="\vec{CA'} = k \cdot \vec{CA}" />.

This means the vector from the center of dilation to the image is <InlineMath math="k" /> times the vector from the center of dilation to the original point.

- If <InlineMath math="|k| > 1" />, it is an enlargement.
- If <InlineMath math="0 < |k| < 1" />, it is a reduction.
- If <InlineMath math="k > 0" />, the original point and its image are on the same side of the center of dilation.
- If <InlineMath math="k < 0" />, the original point and its image are on opposite sides of the center of dilation (and the image is inverted).

## Dilation with Respect to the Origin with Scale Factor k

If the center of dilation is the origin <InlineMath math="O(0,0)" /> and the scale factor is <InlineMath math="k" />, then for a point <InlineMath math="A(x,y)" />, its image <InlineMath math="A'(x',y')" /> is given by:

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

### Dilating a Point with Respect to the Origin

If point <InlineMath math="A(1,2)" /> is dilated with respect to the origin <InlineMath math="(0,0)" /> by a factor of <InlineMath math="2" />, determine the image of the point.

Here, <InlineMath math="x=1" />, <InlineMath math="y=2" />, and <InlineMath math="k=2" />.

The center of dilation is <InlineMath math="O(0,0)" />.

<div className="flex flex-col gap-4">
  <BlockMath math="x' = 2 \cdot 1 = 2" />
  <BlockMath math="y' = 2 \cdot 2 = 4" />
</div>

Thus, the image is <InlineMath math="A'(2,4)" />.

<LineEquation
  title={
    <>
      Dilation of Point <InlineMath math="A(1,2)" /> from Origin, Factor{" "}
      <InlineMath math="k=2" />
    </>
  }
  description={
    <>
      Visualization of dilating point <InlineMath math="A(1,2)" /> to{" "}
      <InlineMath math="A'(2,4)" /> with center at <InlineMath math="O(0,0)" />{" "}
      and scale factor 2.
    </>
  }
  data={[
    {
      points: [{ x: 0, y: 0, z: 0 }],
      color: getColor("ROSE"),
      showPoints: true,
      labels: [{ text: "O(0,0)", at: 0, offset: [0.3, -0.3, 0] }],
    }, // Center of Dilation
    {
      points: [{ x: 1, y: 2, z: 0 }],
      color: getColor("SKY"),
      showPoints: true,
      labels: [{ text: "A(1,2) - Original", at: 0, offset: [1, 0.2, 0] }],
    }, // Original Point
    {
      points: [{ x: 2, y: 4, z: 0 }],
      color: getColor("EMERALD"),
      showPoints: true,
      labels: [{ text: "A'(2,4) - Image", at: 0, offset: [0.3, 0.3, 0] }],
    }, // Image Point
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 1, y: 2, z: 0 },
      ],
      color: getColor("INDIGO"),
    }, // Line OA
    {
      points: [
        { x: 0, y: 0, z: 0 },
        { x: 2, y: 4, z: 0 },
      ],
      color: getColor("INDIGO"),
    }, // Line OA'
  ]}
  showZAxis={false}
  cameraPosition={[1, 2, 15]}
/>

## Dilation with Respect to an Arbitrary Point with Scale Factor k

If the center of dilation is an arbitrary point <InlineMath math="C(a,b)" /> and the scale factor is <InlineMath math="k" />, then for a point <InlineMath math="A(x,y)" />, its image <InlineMath math="A'(x',y')" /> is given by:

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

This can be interpreted as: translate the system so that <InlineMath math="C" /> becomes the origin, perform the dilation by factor <InlineMath math="k" />, and then translate back.

### Dilating a Point with Respect to an Arbitrary Point

If point <InlineMath math="C(5,2)" /> is dilated with respect to point <InlineMath math="P(2,3)" /> by a factor of <InlineMath math="2" />, determine the image of the point.

Here, the point to be dilated is <InlineMath math="C(5,2)" /> so <InlineMath math="x=5, y=2" />.

The center of dilation is <InlineMath math="P(2,3)" />, so <InlineMath math="a=2, b=3" />.

The scale factor is <InlineMath math="k=2" />.

<div className="flex flex-col gap-4">
  <BlockMath math="x' = 2 + 2(5 - 2) = 2 + 2(3) = 2 + 6 = 8" />
  <BlockMath math="y' = 3 + 2(2 - 3) = 3 + 2(-1) = 3 - 2 = 1" />
</div>

Thus, the image is <InlineMath math="C'(8,1)" />.

<LineEquation
  title={
    <>
      Dilation of Point <InlineMath math="C(5,2)" /> from{" "}
      <InlineMath math="P(2,3)" />, Factor <InlineMath math="k=2" />
    </>
  }
  description={
    <>
      Visualization of dilating point <InlineMath math="C(5,2)" /> to{" "}
      <InlineMath math="C'(8,1)" /> with center at <InlineMath math="P(2,3)" />{" "}
      and scale factor 2.
    </>
  }
  data={[
    {
      points: [{ x: 2, y: 3, z: 0 }],
      color: getColor("ROSE"),
      showPoints: true,
      labels: [{ text: "P(2,3) - Center", at: 0, offset: [1, 0.5, 0] }],
    }, // Center of Dilation
    {
      points: [{ x: 5, y: 2, z: 0 }],
      color: getColor("SKY"),
      showPoints: true,
      labels: [{ text: "C(5,2) - Original", at: 0, offset: [0.3, -0.3, 0] }],
    }, // Original Point
    {
      points: [{ x: 8, y: 1, z: 0 }],
      color: getColor("EMERALD"),
      showPoints: true,
      labels: [{ text: "C'(8,1) - Image", at: 0, offset: [0.5, -0.5, 0] }],
    }, // Image Point
    {
      points: [
        { x: 2, y: 3, z: 0 },
        { x: 5, y: 2, z: 0 },
      ],
      color: getColor("INDIGO"),
    }, // Line PC
    {
      points: [
        { x: 2, y: 3, z: 0 },
        { x: 8, y: 1, z: 0 },
      ],
      color: getColor("INDIGO"),
    }, // Line PC'
  ]}
  showZAxis={false}
/>

## Exercises

1.  Determine the image of <InlineMath math="B(2,5)" /> under dilation <InlineMath math="D_{(O,3)}" /> (center at O(0,0), factor 3).
2.  Determine the image of <InlineMath math="B(2,5)" /> under dilation with center <InlineMath math="P(1,3)" /> and factor 3.
3.  A triangle with vertices <InlineMath math="A(1,1)" />, <InlineMath math="B(3,1)" />, and <InlineMath math="C(1,4)" /> is dilated with respect to the origin <InlineMath math="O(0,0)" /> by a scale factor <InlineMath math="k=-2" />. Determine the coordinates of the image triangle <InlineMath math="A'B'C'" />!

### Key Answers

1.  Point <InlineMath math="B(2,5)" />, center <InlineMath math="O(0,0)" />, <InlineMath math="k=3" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="x' = 3 \cdot 2 = 6" />
      <BlockMath math="y' = 3 \cdot 5 = 15" />
    </div>

    Thus, the image is <InlineMath math="B'(6,15)" />.

2.  Point <InlineMath math="B(2,5)" />, center <InlineMath math="P(1,3)" />, <InlineMath math="k=3" />. (<InlineMath math="x=2, y=5, a=1, b=3" />)

    <div className="flex flex-col gap-4">
      <BlockMath math="x' = 1 + 3(2 - 1) = 1 + 3(1) = 1 + 3 = 4" />
      <BlockMath math="y' = 3 + 3(5 - 3) = 3 + 3(2) = 3 + 6 = 9" />
    </div>

    Thus, the image is <InlineMath math="B'(4,9)" />.

3.  Center <InlineMath math="O(0,0)" />, <InlineMath math="k=-2" />.

    - For <InlineMath math="A(1,1)" />: <InlineMath math="A'(-2 \cdot 1, -2 \cdot 1) = A'(-2,-2)" />.
    - For <InlineMath math="B(3,1)" />: <InlineMath math="B'(-2 \cdot 3, -2 \cdot 1) = B'(-6,-2)" />.
    - For <InlineMath math="C(1,4)" />: <InlineMath math="C'(-2 \cdot 1, -2 \cdot 4) = C'(-2,-8)" />.

    The coordinates of the image triangle are: <InlineMath math="A'(-2,-2)" />, <InlineMath math="B'(-6,-2)" />, <InlineMath math="C'(-2,-8)" />.