Similarity transformations and dilations
Similarity Transformations and Dilations
Similarity Transformations and Dilations is the opening topic of Similarity in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.7(A), which requires students to apply the definition of similarity in terms of a dilation.
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.