Similarity Transformation — Definition, Formula & Examples

A similarity transformation is a sequence of rigid motions (translations, reflections, rotations) and dilations that maps one figure onto a similar figure. The resulting image has the same shape as the original but may differ in size, position, or orientation.

A similarity transformation is a composition of a dilation with scale factor $k \neq 0$ and one or more isometries (distance-preserving transformations) such that for any two points $A$ and $B$ in the pre-image and their images $A'$ and $B'$ , the relationship $A'B' = |k| \cdot AB$ holds. Two figures are similar if and only if one can be mapped onto the other by a similarity transformation.

How It Works

To perform a similarity transformation, you apply a dilation and one or more rigid motions in sequence. The order can vary — you might dilate first, then rotate, or reflect first, then dilate. The dilation changes the size of the figure by a scale factor

k

, while the rigid motions reposition or reorient it without changing its size. All corresponding angles remain congruent, and all corresponding side lengths share the same ratio

|k|

Worked Example

Problem: Triangle

ABC

has vertices

A(0,0)

B(4,0)

, and

C(0,3)

. Apply a similarity transformation consisting of a dilation centered at the origin with scale factor

k = 2

, followed by a reflection over the

x

-axis. Find the vertices of the image triangle.

Step 1: Apply the dilation: Multiply each coordinate by the scale factor 2.

A' = (0,0), \quad B' = (8,0), \quad C' = (0,6)

Step 2: Reflect over the x-axis: Negate each

y

-coordinate to reflect across the

x

-axis.

A'' = (0,0), \quad B'' = (8,0), \quad C'' = (0,-6)

Step 3: Verify similarity: The original triangle has sides 4, 3, and 5. The image triangle has sides 8, 6, and 10 — each exactly twice the original. All angles are preserved.

\frac{A''B''}{AB} = \frac{B''C''}{BC} = \frac{A''C''}{AC} = 2

Answer: The image triangle has vertices

A''(0,0)

B''(8,0)

, and

C''(0,-6)

. It is similar to

\triangle ABC

with a scale factor of 2.

Why It Matters

Similarity transformations are central to proving that two figures are similar in high school geometry, particularly in the Common Core standard approach. Architects and engineers use them when creating scale models, where every measurement must maintain the same proportion to the original structure.

Common Mistakes

Mistake: Assuming a similarity transformation must include a dilation that changes size.

Correction: A dilation with scale factor

k = 1

is still a valid dilation. This means every congruence transformation (rigid motion alone) is also a similarity transformation — congruent figures are a special case of similar figures.