Similarity transformations and dilations
Similarity Transformations and Dilations
Similarity Transformations and Dilations is the opening topic of Similarity in the Common Core State Standards. It is aligned to Standard G-SRT.1, which requires students to verify the properties of dilations and define similarity in terms of similarity transformations.
A dilation scales a figure by a factor \(k\) about a center point, producing a similar figure with proportional lengths and equal angles.
Theory
A dilation resizes a figure from a fixed center by a scale factor \(k\): every point moves to \(k\) times its distance from the center.
- \(k>1\): an enlargement.
- \(0<k<1\): a reduction.
A dilation keeps all angles equal and multiplies all lengths by \(k\), so the image is similar to the original. A similarity transformation is a dilation followed by rigid motions.
Dilation from the origin, and the scale factor:
How to perform a dilation
- Identify the center and scale factor \(k\).
- Multiply each point's distance from the center by \(k\).
- Classify: \(k>1\) enlarges, \(0<k<1\) reduces.
- Conclude the image is similar to the original.
Multiply each coordinate by the scale factor.
| \(A'\) | \(=\) | \((2\cdot 3,\ 2\cdot 4)\) |
| \(=\) | \((6,8)\) |
A factor between 0 and 1 shrinks the figure — a reduction.
| \(0<\dfrac{1}{3}<1\) | \(\Rightarrow\) | \(\text{reduction}\) |
The scale factor is the ratio of image to original.
| \(k\) | \(=\) | \(\dfrac{15}{5}=3\) |
A dilation multiplies every length by the same factor and keeps all angles unchanged, so the shape is preserved — the figures are similar.
Common pitfalls
Frequently asked questions
What is a dilation?
A transformation that resizes a figure from a center point by a scale factor, keeping angles equal.
What does the scale factor tell you?
How much the figure is resized: \(k>1\) enlarges, \(0<k<1\) reduces, and it equals image length over original length.
Is a dilated figure congruent to the original?
Only if \(k=1\). Otherwise it is similar — same shape, different size.
What is a similarity transformation?
A dilation combined with rigid motions; two figures are similar exactly when one maps to the other this way.