# Similarity Transformations

## All Circles Are Similar to the Unit Circle at the Origin...

Every circle can be translated so that its circle is at the origin, and then dilated to a radius of 1. Since translations and dilations are both similarity transformations, this means that every circle is similar to the unit circle at the origin. But then since similarity is transitive, this actually means that all circles are similar to each other.

## 1. Translation

Use the GeoGebra translation tool to translate the circle so that it is centered at the origin. What is the algebraic rule for this translation?

## 2. Image after Translation

Let's call the points on the translated image circle (x', y'). Write equations for x' and y' in terms of the original coordinates x and y.

## 3. Dilation

Use the GeoGebra dilation tool to dilate the translated image circle so that its radius is 1. What is the algebraic rule for this dilation?

## 4. Image after Translation and Dilation

Let's call the points on the translated and dilated image circle (x'',y''). Write equations for x'' and y'' in terms of the original coordinates x and y.