Can a dilation have a scale factor of 0?

No, not in standard geometry. If k = 0, every point would map to the origin, and you'd lose the figure entirely — it would collapse to a single point. Mathematically, this isn't a useful transformation for studying similarity. We always use positive scale factors: k > 0.

If I dilate a figure with scale factor 2, does the area also double?

No, the area is multiplied by k^2, not k. If k = 2, the area becomes 2^2 = 4 times larger. If k = 0.5, the area becomes (0.5)^2 = 0.25 times the original, or one-quarter as large. This is because area is a two-dimensional measurement, so the scale factor affects both dimensions.

What's the difference between dilation and similarity?

A dilation is a specific type of transformation — you apply a scale factor to shrink or enlarge a figure. Similarity is a relationship between two figures: they're similar if one can be obtained from the other by dilation followed by rigid motions. Dilation is the action; similarity is the result or property.

Does the center of dilation have to be at the origin?

In this course, we always dilate from the origin. Dilations can happen from any center point, but when the center is at the origin, the rule is simple: (x, y) → (kx, ky). If the center were elsewhere, the rule would be more complicated, so we focus on the origin.

M8MATH-8.4

Dilations & Similarity

Learn how dilations scale figures from the origin, find scale factors between similar figures, and understand why similarity means dilation plus rigid motions.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Dilations & Similarity, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

You've learned how to slide, flip, and rotate figures without changing their size or shape — those are rigid motions. Now you're about to discover dilations, a completely different kind of transformation that stretches or shrinks a figure. Dilations are how we create similar figures: shapes that have the same form but different sizes. In this lesson, you'll learn to dilate figures using a scale factor, find the scale factor between two figures, and explain what it means for two shapes to be similar.

What Is a Dilation?

A dilation is a transformation that stretches or shrinks a figure from a fixed point, called the center of dilation. When we dilate from the origin with scale factor

k

, every point

(x, y)

on the figure moves to a new point

(kx, ky)

. If

k > 1

, the figure gets larger. If

0 < k < 1

, the figure gets smaller. If

k = 1

, nothing changes — that's the identity dilation.

Dilations preserve shape but not size. All angles stay the same, and the ratios of side lengths stay the same. For example, if one side is twice as long as another, that ratio holds after dilation. But unless

k = 1

, the actual side lengths change by a factor of

k

.

Unlike translations, reflections, and rotations (the rigid motions), dilations are not congruence transformations. Two figures related by a dilation alone are not congruent — they're similar.

Finding the Scale Factor

When you're given two figures and told one is a dilation of the other, you can find the scale factor

k

by comparing corresponding side lengths. If a side on the original figure has length

\ell

and the same side on the dilated figure has length

\ell'

, then

k = \frac{\ell'}{\ell}

.

For example, suppose triangle

ABC

has a side of length 4 units. After dilation from the origin, the corresponding side of triangle

A'B'C'

has length 10 units. Then

k = \frac{10}{4} = 2.5

.

You can also find

k

by comparing coordinates. If point

A = (2, 3)

dilates to

A' = (6, 9)

, then

k = \frac{6}{2} = 3

k = \frac{9}{3} = 3

. Both ratios must match. If they don't, the transformation wasn't a dilation from the origin.

A common mistake is to confuse the ratio the other way: using

k = \frac{\ell}{\ell'}

instead of

k = \frac{\ell'}{\ell}

. Always put the new figure's measurement in the numerator.

Similarity Through Dilation and Rigid Motions

Two figures are similar if one can be obtained from the other through a sequence of transformations: first a dilation, then any combination of rigid motions (translations, reflections, rotations).

For instance, if square

ABCD

with side length 2 is dilated by scale factor 3 from the origin, you get square

A'B'C'D'

with side length 6. These two squares are similar just from the dilation. But you could also rotate

A'B'C'D'

and then translate it, and the squares would still be similar to

ABCD

. The order matters: dilation first, then rigid motions.

Why this definition? Rigid motions preserve both shape and size, so they don't change similarity. Dilations change size but preserve shape. Together, they capture exactly what similarity means: same shape, possibly different size.

This is different from congruence. Two figures are congruent if they're related by rigid motions alone, with no dilation. Congruent figures are always similar (with

k = 1

), but similar figures are not always congruent.

Working with Dilations from the Origin

To dilate a figure from the origin with scale factor

k

:

Identify all vertices of the original figure. For each vertex

(x, y)

, multiply both coordinates by

k

to get the new vertex

(kx, ky)

. Connect the new vertices in the same order as the original.

Example: triangle with vertices

A = (1, 2)

B = (4, 2)

C = (2, 5)

, dilated by

k = 2

from the origin.

New vertices:

A' = (2, 4)

B' = (8, 4)

C' = (4, 10)

.

Notice that side

AB

originally goes from

(1, 2)

(4, 2)

, with length 3. After dilation,

A'B'

goes from

(2, 4)

(8, 4)

, with length 6. The side length doubled, as expected when

k = 2

.

For scale factors between 0 and 1, the figure shrinks toward the origin. If

k = 0.5

and a point is at

(8, 6)

, it moves to

(4, 3)

— closer to the origin. The shape stays the same, just smaller.

Key terms

Dilation.: A transformation that stretches or shrinks a figure from a center of dilation by a scale factor $k$ , where each point $(x, y)$ maps to $(kx, ky)$ when the center is at the origin.
Scale factor.: The ratio $k$ that tells you how much larger or smaller a dilated figure is compared to the original; $k > 1$ enlarges, $0 < k < 1$ shrinks, and $k = 1$ leaves it unchanged.
Center of dilation.: The fixed point from which a dilation occurs; in this lesson, always the origin $(0, 0)$ .
Similar figures.: Two figures with the same shape but possibly different sizes; one is obtained from the other through a dilation followed by any rigid motions.
Corresponding sides.: Sides of two similar figures that match up in position and order; they are proportional, with the ratio equal to the scale factor.
Congruent figures.: Two figures with the same shape and size, related by rigid motions only (no dilation).
Rigid motion.: A transformation (translation, reflection, or rotation) that preserves both the size and shape of a figure.

Worked example

Triangle

PQR

has vertices

P = (2, 1)

Q = (6, 1)

, and

R = (4, 5)

. Dilate this triangle from the origin with scale factor

k = 1.5

. Then find the length of side

P'Q'

and compare it to side

PQ

Step 1: Dilate each vertex by multiplying coordinates by 1.5.

P = (2, 1) \to P' = (1.5 \times 2, 1.5 \times 1) = (3, 1.5)

Q = (6, 1) \to Q' = (1.5 \times 6, 1.5 \times 1) = (9, 1.5)

R = (4, 5) \to R' = (1.5 \times 4, 1.5 \times 5) = (6, 7.5)

Step 2: Find the length of the original side

PQ

. Since

P = (2, 1)

and

Q = (6, 1)

have the same

y

-coordinate, they form a horizontal segment. The length is

|6 - 2| = 4

units.

Step 3: Find the length of the dilated side

P'Q'

. Since

P' = (3, 1.5)

and

Q' = (9, 1.5)

also have the same

y

-coordinate, the length is

|9 - 3| = 6

units.

Step 4: Compare. The dilated side is

\frac{6}{4} = 1.5

times the original side, which matches our scale factor

k = 1.5

. The triangle

P'Q'R'

is similar to

PQR

, enlarged by 50 percent.

Practice questions

Point

A = (4, 3)

is dilated from the origin with scale factor

k = 0.5

to get point

A'

. What are the coordinates of

A'

(2, 1.5)
(8, 6)
(2, 3)
(4, 1.5)

Answer: (2, 1.5)

Using the dilation rule

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

with

k = 0.5

A' = (0.5 \times 4, 0.5 \times 3) = (2, 1.5)

. This scale factor shrinks the point toward the origin. A common mistake is doubling the coordinates instead; that would give

(8, 6)

, which is wrong.

Rectangle

ABCD

has vertices at

A = (1, 2)

B = (5, 2)

C = (5, 6)

, and

D = (1, 6)

. After a dilation from the origin, the image rectangle

A'B'C'D'

has vertices at

A' = (3, 6)

B' = (15, 6)

C' = (15, 18)

, and

D' = (3, 18)

. Find the scale factor.

Answer: 3

Compare corresponding side lengths. Side

AB

goes from

(1, 2)

(5, 2)

, with length 4. Side

A'B'

goes from

(3, 6)

(15, 6)

, with length 12. The scale factor is

k = \frac{12}{4} = 3

. You can verify with another pair:

AD

has length

6 - 2 = 4

, and

A'D'

has length

18 - 6 = 12

, giving

k = \frac{12}{4} = 3

again.

Explain why two figures that are related by a dilation alone (with scale factor

k \neq 1

) are similar but not congruent.

Answer: A dilation preserves the shape of a figure — all angles stay the same and side lengths scale proportionally — but it changes the size unless $k = 1$ . Congruence requires both the same shape and the same size, which is why the figures must be related by rigid motions only. A dilation changes size, so figures related by a dilation (and no rigid motions) cannot be congruent. However, they are similar because similarity allows for a change in size as long as the shape is preserved.

This question tests understanding of the definitions. The key insight is that congruence and similarity differ in whether size is preserved. Congruent figures have the same size and shape (rigid motions only). Similar figures have the same shape but possibly different sizes (dilation allowed). Dilations preserve shape but not size, so they create similar but not congruent figures.

FAQ

Can a dilation have a scale factor of 0?: No, not in standard geometry. If $k = 0$ , every point would map to the origin, and you'd lose the figure entirely — it would collapse to a single point. Mathematically, this isn't a useful transformation for studying similarity. We always use positive scale factors: $k > 0$ .
If I dilate a figure with scale factor 2, does the area also double?: No, the area is multiplied by $k^2$ , not $k$ . If $k = 2$ , the area becomes $2^2 = 4$ times larger. If $k = 0.5$ , the area becomes $(0.5)^2 = 0.25$ times the original, or one-quarter as large. This is because area is a two-dimensional measurement, so the scale factor affects both dimensions.
What's the difference between dilation and similarity?: A dilation is a specific type of transformation — you apply a scale factor to shrink or enlarge a figure. Similarity is a relationship between two figures: they're similar if one can be obtained from the other by dilation followed by rigid motions. Dilation is the action; similarity is the result or property.
Does the center of dilation have to be at the origin?: In this course, we always dilate from the origin. Dilations can happen from any center point, but when the center is at the origin, the rule is simple: $(x, y) \to (kx, ky)$ . If the center were elsewhere, the rule would be more complicated, so we focus on the origin.

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