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URL: https://nakafa.com/en/subjects/mathematics/geometric-transformation/dilation
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Learn geometric dilation with scale factors and center points. Learn enlargement & reduction formulas with visual examples and practice problems.

---

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

Visible text: Given a point as the center of dilation and a scale factor . The dilation of a point with respect to center by a factor , denoted as , is a transformation that maps 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 $$k\text{ times}$$ the vector from the center of dilation to the original point.

Visible text: This means the vector from the center of dilation to the image is the vector from the center of dilation to the original point.

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

Visible text: - If , it is an enlargement.
- If , it is a reduction.
- If , the original point and its image are on the same side of the center of dilation.
- If , 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

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

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

Component: MathContainer
Children:

```math
x' = kx
```

```math
y' = ky
```

### Dilating a Point with Respect to the Origin

If point $$A(1,2)$$ is dilated with respect to the origin $$(0,0)$$ by a factor of $$2$$, determine the image of the point.

Visible text: If point is dilated with respect to the origin by a factor of , determine the image of the point.

Here, $$x=1$$, $$y=2$$, and $$k=2$$.

Visible text: Here, , , and .

The center of dilation is $$O(0,0)$$.

Visible text: The center of dilation is .

Component: MathContainer
Children:

```math
x' = 2 \cdot 1 = 2
```

```math
y' = 2 \cdot 2 = 4
```

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

Component: LineEquation
Props:
- title: Dilation of Point $$A(1,2)$$ from Origin, Factor{" "}
$$k=2$$
  Visible text: Dilation of Point from Origin, Factor{" "}
- description: Visualization of dilating point $$A(1,2)$$ to{" "}
<InlineMath math="A'(2,4)" /> with center at $$O(0,0)$${" "}
and scale factor $$2$$.
  Visible text: Visualization of dilating point to{" "}
<InlineMath math="A'(2,4)" /> with center at {" "}
and scale factor .
- 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

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

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

Component: MathContainer
Children:

```math
x' = a + k(x - a)
```

```math
y' = b + k(y - b)
```

This can be interpreted as: translate the system so that $$C$$ becomes the origin, perform the dilation by factor $$k$$, and then translate back.

Visible text: This can be interpreted as: translate the system so that becomes the origin, perform the dilation by factor , and then translate back.

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

If point $$C(5,2)$$ is dilated with respect to point $$P(2,3)$$ by a factor of $$2$$, determine the image of the point.

Visible text: If point is dilated with respect to point by a factor of , determine the image of the point.

Here, the point to be dilated is $$C(5,2)$$ so $$x=5, y=2$$.

Visible text: Here, the point to be dilated is so .

The center of dilation is $$P(2,3)$$, so $$a=2, b=3$$.

Visible text: The center of dilation is , so .

The scale factor is $$k=2$$.

Visible text: The scale factor is .

Component: MathContainer
Children:

```math
x' = 2 + 2(5 - 2) = 2 + 2(3) = 2 + 6 = 8
```

```math
y' = 3 + 2(2 - 3) = 3 + 2(-1) = 3 - 2 = 1
```

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

Component: LineEquation
Props:
- title: Dilation of Point $$C(5,2)$$ from{" "}
$$P(2,3)$$, Factor $$k=2$$
  Visible text: Dilation of Point from{" "}
, Factor
- description: Visualization of dilating point $$C(5,2)$$ to{" "}
<InlineMath math="C'(8,1)" /> with center at $$P(2,3)$${" "}
and scale factor $$2$$.
  Visible text: Visualization of dilating point to{" "}
<InlineMath math="C'(8,1)" /> with center at {" "}
and scale factor .
- 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 $$B(2,5)$$ under dilation $$D_{(O,3)}$$ (center at $$O(0,0)$$, factor $$3$$).
2.  Determine the image of $$B(2,5)$$ under dilation with center $$P(1,3)$$ and factor $$3$$.
3.  A triangle with vertices $$A(1,1)$$, $$B(3,1)$$, and $$C(1,4)$$ is dilated with respect to the origin $$O(0,0)$$ by a scale factor $$k=-2$$. Determine the coordinates of the image triangle <InlineMath math="A'B'C'" />!

Visible text: 1. Determine the image of under dilation (center at , factor ).
2. Determine the image of under dilation with center and factor .
3. A triangle with vertices , , and is dilated with respect to the origin by a scale factor . Determine the coordinates of the image triangle <InlineMath math="A'B'C'" />!

### Key Answers

1.  Point $$B(2,5)$$, center $$O(0,0)$$, $$k=3$$.

    <MathContainer>
      <BlockMath math="x' = 3 \cdot 2 = 6" />
      <BlockMath math="y' = 3 \cdot 5 = 15" />
    </MathContainer>

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

2.  Point $$B(2,5)$$, center $$P(1,3)$$, $$k=3$$. ($$x=2, y=5, a=1, b=3$$)

    <MathContainer>
      <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" />
    </MathContainer>

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

3.  Center $$O(0,0)$$, $$k=-2$$.

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

Visible text: 1. Point , center , .

 <MathContainer>
 <BlockMath math="x' = 3 \cdot 2 = 6" />
 <BlockMath math="y' = 3 \cdot 5 = 15" />
 </MathContainer>

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

2. Point , center , . ()

 <MathContainer>
 <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" />
 </MathContainer>

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

3. Center , .

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