Dilations

Q: Is the image of a dilation congruent to the original?

Not unless $k = 1$. A dilation produces a similar figure — the angles match and the shape is identical, but every side length is multiplied by $k$, so the sizes differ.

Q: What does "centered at the origin" mean?

The origin is the fixed point the figure grows away from or shrinks toward. Points on the image lie along the same lines through $(0, 0)$ as their originals, just $k$ times as far out.

Q: What if the scale factor is a fraction?

The rule is the same — multiply both coordinates by it. A fraction between $0$ and $1$ shrinks the figure: scale factor $\dfrac{1}{4}$ sends $(8, 12)$ to $(2, 3)$.

Q: How is a dilation different from the other transformations?

Translations, reflections, and rotations move a figure without resizing it, so their images are congruent. A dilation is the only one of the four that changes size, so its image is similar instead.

A dilation makes a figure larger or smaller without changing its shape. It is the odd one out among the transformations: translations, reflections, and rotations all keep the figure the same size, but a dilation resizes it. The image is similar to the original — same shape, same angles, different size.

On the coordinate plane, a dilation centered at the origin is one multiplication: multiply both coordinates of every point by the scale factor $k$ . That single number controls everything about how the figure grows or shrinks.

The dilation rule

A dilation centered at the origin with scale factor $k$ uses the rule $(x, y) \to (kx, ky)$ . Multiply the $x$ -coordinate by $k$ and the $y$ -coordinate by $k$ — both of them, every point.

The size of $k$ tells you what happens. If $k > 1$ , the image is larger than the original (an enlargement). If $0 < k < 1$ , the image is smaller (a reduction). If $k = 1$ , nothing changes at all.

Because both coordinates are multiplied by the same number, every point of the image is $k$ times as far from the origin as its original — the figure grows or shrinks away from or toward $(0, 0)$ .

Finding the scale factor from a graph

When you are given a figure and its dilated image, pick one vertex of the original and its matching image vertex, then divide: an image coordinate divided by the matching original coordinate gives $k$ .

In the graph below, the small black triangle has a corner at $(2, 1)$ and the blue image has its matching corner at $(4, 2)$ . Dividing, $4 \div 2 = 2$ and $2 \div 1 = 2$ , so the scale factor is $k = 2$ . Both divisions must give the same number — if they disagree, the figures are not dilations of each other, or you matched the wrong vertices.

Multiply — never add

The most common dilation mistake is adding the scale factor instead of multiplying by it. A scale factor of $4$ applied to $(-2, 3)$ gives $(-8, 12)$ , not $(2, 7)$ . If your answer came from adding, it is a translation, not a dilation.

Fractions work the same way. A scale factor of $\dfrac{1}{2}$ means multiply each coordinate by $\dfrac{1}{2}$ — which is the same as dividing each coordinate by $2$ .

Worked examples

Example 1: an enlargement

A dilation centered at the origin has scale factor $3$ . Find the image of $(2, -5)$ .

Write the rule with

k = 3

(x, y) \to (3x, 3y)

Multiply the

x

-coordinate by

3

3 \cdot 2 = 6

Multiply the

y

-coordinate by

3

3 \cdot (-5) = -15

Answer: $(6, -15)$

Example 2: a reduction

A dilation centered at the origin has scale factor $\dfrac{1}{2}$ . Find the image of $(8, -4)$ .

Write the rule with

k = \dfrac{1}{2}

(x, y) \to \left(\tfrac{1}{2}x, \tfrac{1}{2}y\right)

Multiply the

x

-coordinate by

\dfrac{1}{2}

\tfrac{1}{2} \cdot 8 = 4

Multiply the

y

-coordinate by

\dfrac{1}{2}

\tfrac{1}{2} \cdot (-4) = -2

Answer: $(4, -2)$

Example 3: find the scale factor

A dilation centered at the origin sends $B(2, 1)$ to $B'(6, 3)$ . What is the scale factor?

Write the rule

(x, y) \to (kx, ky)

Divide an image coordinate by its original

6 \div 2 = 3

Check with the other coordinate

3 \div 1 = 3

Answer: $k = 3$

Try one yourself

Common questions

Is the image of a dilation congruent to the original?

Not unless $k = 1$ . A dilation produces a similar figure — the angles match and the shape is identical, but every side length is multiplied by $k$ , so the sizes differ.

What does "centered at the origin" mean?

The origin is the fixed point the figure grows away from or shrinks toward. Points on the image lie along the same lines through $(0, 0)$ as their originals, just $k$ times as far out.

What if the scale factor is a fraction?

The rule is the same — multiply both coordinates by it. A fraction between $0$ and $1$ shrinks the figure: scale factor $\dfrac{1}{4}$ sends $(8, 12)$ to $(2, 3)$ .

How is a dilation different from the other transformations?

Translations, reflections, and rotations move a figure without resizing it, so their images are congruent. A dilation is the only one of the four that changes size, so its image is similar instead.

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Dilations

The dilation rule

Finding the scale factor from a graph

Multiply — never add

Worked examples

Example 1: an enlargement

Example 2: a reduction

Example 3: find the scale factor

Try one yourself

Common questions

Keep learning

Translations

Reflections

Rotations

AA Similarity