Dilations and Similarity Transformations

In geometry, transformations change the position, size, or shape of a figure. While rigid motions (like translations and rotations) keep the size of a figure exactly the same, dilations change the size of a figure without changing its shape.

What is a Dilation?

A dilation is a transformation that stretches or shrinks a figure. Every dilation requires two things:

1. Center of Dilation: A fixed point from which all points are expanded or contracted.
2. Scale Factor ($k$): The ratio of a length in the image (the new figure) to the corresponding length in the pre-image (the original figure).

• If $k > 1$, the dilation is an enlargement.
• If $0 < k < 1$, the dilation is a reduction.
• If $k = 1$, the figure stays the exact same size (identity transformation).

Key Properties of Dilations:

• Angle measures are preserved. The angles of the original figure and the dilated figure are identical.
• Lengths are changed proportionally. Every side length is multiplied by the scale factor $k$.
• Because angles are preserved and sides are proportional, a figure and its dilated image are always similar.

Dilations in the Coordinate Plane

When the center of dilation is the origin $(0,0)$, finding the coordinates of the dilated image is straightforward. You simply multiply the $x$ and $y$ coordinates of each vertex by the scale factor $k$: $(x, y) \rightarrow (kx, ky)$

Example: Dilation at the Origin

Problem: Find the image of $\triangle ABC$ with vertices $A(2,4)$, $B(6,2)$, and $C(4,8)$ under a dilation with center $O(0,0)$ and a scale factor of $k = \frac{1}{2}$.

Solution: Apply the rule $(x, y) \rightarrow \left(\frac{1}{2}x, \frac{1}{2}y\right)$ to each vertex:

• $A(2, 4) \rightarrow A'(1, 2)$
• $B(6, 2) \rightarrow B'(3, 1)$
• $C(4, 8) \rightarrow C'(2, 4)$

The new vertices form $\triangle A'B'C'$, which is exactly half the size of the original triangle but identical in shape.

Similarity Transformations

A similarity transformation is a combination of one or more rigid motions (translations, reflections, or rotations) followed by a dilation.

While rigid motions produce congruent figures (same shape and size), similarity transformations produce similar figures (same shape, different size). If you can map one figure onto another using a combination of rotations, reflections, translations, and dilations, the two figures are mathematically similar.

Finding the Scale Factor and Center

If you are given a figure and its dilated image, you can work backward to find the scale factor and the center of dilation.

• To find the scale factor ($k$): Pick any side on the image and divide its length by the corresponding side length on the pre-image. $k = \frac{\text{Length of Image}}{\text{Length of Pre-image}}$
• To find the center of dilation: Draw straight lines connecting corresponding vertices of the image and pre-image (e.g., a line through $A$ and $A'$, another through $B$ and $B'$). The point where all these lines intersect is the center of dilation.