9.2 Dilations and similarity transformations

Dilations

more resources to help you study

practice questions

Dilations in the Coordinate Plane

A dilation is a transformation that changes a figure's size without changing its shape. Every dilation is defined by two things: a center of dilation (the fixed point everything expands from or shrinks toward) and a scale factor that controls how much the figure grows or shrinks.

The scale factor $k$ tells you exactly what happens:

• $k > 1$: the figure enlarges (e.g., $k = 3$ triples every distance from the center)
• $0 < k < 1$: the figure shrinks (e.g., $k = \frac{1}{2}$ cuts every distance in half)
• $k < 0$: the figure is dilated and reflected through the center (the image ends up on the opposite side)

One detail worth remembering: the center of dilation itself never moves. It's the one point that maps to itself under any dilation. The center can be the origin, a vertex of the figure, or any other point in the plane. Where it's located affects how you compute the new coordinates.

Scale Factors in Dilations

When the center of dilation is the origin, you multiply each coordinate by the scale factor $k$:

This works because every distance from the center scales by the same factor. If a vertex is 2 units from the center and $k = 3$, the image of that vertex is $2 \times 3 = 6$ units from the center. The absolute value of $k$, written $|k|$, gives you the ratio of image distance to original distance. So when $k = -3$, the image is still 6 units from the center, just on the opposite side.

Coordinates After Dilation

Center at the origin: Multiply directly.

• Point $(2, 3)$ with $k = 2$ maps to $(4, 6)$.
• Point $(-1, 4)$ with $k = \frac{1}{2}$ maps to $(-\frac{1}{2}, 2)$.

Center NOT at the origin: You need three steps. The idea is to temporarily reposition everything so the center sits at the origin, perform the dilation there, then shift everything back.

1. Translate so the center of dilation moves to the origin. Subtract the center's coordinates from each point.
2. Dilate by multiplying the translated coordinates by $k$.
3. Translate back by adding the center's coordinates to the result.

You can also express this as a single formula. For a point $(x, y)$ dilated by factor $k$ about center $(a, b)$:

For example, dilate point $(5, 7)$ by $k = 2$ with center $(1, 3)$:

1. Translate: $(5 - 1,\; 7 - 3) = (4, 4)$

2. Dilate: $(2 \cdot 4,\; 2 \cdot 4) = (8, 8)$

3. Translate back: $(8 + 1,\; 8 + 3) = (9, 11)$

Using the formula directly: $(2(5-1)+1,\; 2(7-3)+3) = (9, 11)$. Same answer.

Similarity Transformations

Dilations vs. Similarity Transformations

A similarity transformation is any sequence of transformations that produces a figure with the same shape as the original. Dilations are one type, but translations, rotations, and reflections also play a role. The difference is that translations, rotations, and reflections are rigid motions (they preserve both shape and size), while dilations change size.

Here's how they fit together: a similarity transformation is a dilation combined with zero or more rigid motions. If you dilate a figure and then slide, rotate, or reflect it, the final image is similar to the original. A rigid motion alone is actually a special case of a similarity transformation where $k = 1$.

The order of operations can matter for computing coordinates, but the end result is always a similar figure regardless of whether you dilate first or apply the rigid motion first.

Similar Figures Through Dilations

Two figures are similar (written $\sim$) when they have the same shape but not necessarily the same size. That requires two conditions:

• Corresponding angles are congruent (equal in measure).
• Corresponding sides are proportional (all share the same ratio).

To create a similar figure, dilate the original by some scale factor $k$, then optionally apply any rigid motion. To verify that two figures are similar, check whether corresponding sides all share a common ratio and corresponding angles are equal.

For example, a triangle with sides 3, 4, 5 and a triangle with sides 6, 8, 10:

$\frac{6}{3} = \frac{8}{4} = \frac{10}{5} = 2$

Every pair of corresponding sides has the same ratio, so the triangles are similar with a scale factor of $k = 2$. That ratio of corresponding side lengths is the scale factor of the dilation that maps one figure onto the other.

A common mistake: make sure you're matching the right pairs of sides. Always pair the shortest side with the shortest side, the longest with the longest (or use angle correspondence if you have it). If you mismatch the pairing, the ratios won't come out equal and you might wrongly conclude the figures aren't similar.