# Similarity Transformations Worksheets

We have been discussing the concept of various transformations where a geometric figure or shape changes in some way normally on a coordinate plane. Now it is time to diagnose a possible change or movement of a shape. A similarity transformation is when either a rotation, reflection, or translation occurs and then the shape changes scale or size via a dilation. To do this you will you need to track the possible trajectory that this shape could have taken by reverse engineering it. Once you declare that it is possible, you will check your work by verify that the corresponding angles are congruent and that the sides are in direct proportion to one another. These worksheets and lessons help students learn how to recognize and perform similarity transformation which are standard transformations that are followed by a dilation.

### Aligned Standard: High School Geometry - HSG-SRT.A.2

- Changing Triangles Step-by-step Lesson - How does the one triangle become the other triangle?
- Guided Lesson - Determine the rules that dictate the size of these triangles and other shapes.
- Guided Lesson Explanation - These explanations are not for the weak reader. I need to find a way to make them more graphical.
- Practice Worksheet - A five pack to test your skills and endurance.
- Matching Worksheet - Ya, you really only need to do one problem. If you do both, you can check your work that way.
- Intuitive Notion of Translations Worksheet Five Pack - Is translation at work? The second step is to determine if a horizontal or vertical translation took place.
- Translations Worksheet Five Pack - A nice true or false starts you off. Then we move to graphs.
- Transformation-Dilation Worksheet Five Pack - Determine some scalar factors for us.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

This skill is somewhat the inverse of the previous standard. It just requires you to think at a slightly higher level.

- Homework 1 - After translating the center of F to the center of F', dilate F about its new center to contract F onto F'.
- Homework 2 - Choose a pair of corresponding vertices not located at the origin, like V' (2, -2) and the image of V after the translation, which is V' (10,-10).
- Homework 3 - To find the scale factor of this dilation, calculate the ratio of the radii.

### Practice Worksheets

If students are having trouble start with the second version here.

- Practice 1 - You can transform triangle PQR to triangle P'Q'R' by translating it and then performing dilation centered at the origin. So, triangle PQR ~ triangle P'Q'R'. Find the translation rule and the scale factor of the dilation.
- Practice 2 - You can transform S to S' by translating it and then performing a dilation. Find the translation rule and the scale factor of the dilation.
- Practice 3 - See if you can model what is asked for.

### Math Skill Quizzes

I mix some fun questions with problems that require a little bit of thinking.

- Quiz 1 - Determine if it is an example of translation?
- Quiz 2 - Under a translation of 2 units up and 4 units to the left, the point (6, 6) will become (-10,-3). Is this translation is true or false?
- Quiz 3 - If the length of the plastic cover is 16 inches under the scale factor of the dilation is 4. What is the width of the plastic cover after the dilation?

### What are the Similarity Transformations?

Transformation in mathematics is a fun concept as it allows moving an object from its original position and orientation to a new position and/or orientation. There are three types of transformations that we can perform on geometrical shapes, including reflection, rotation, and translation. We know what transformations are, but do we know what similarity transformations are?

A similarity transformation is a case when the resulting image after applying one or more rigid transformations is similar to the original figure. A rigid transformation is applied to the figure and a dilation then follows it. The resulting image has the same angles and all the sides of the original figure, and the images are proportional.

If you want to determine if two geometric shapes are similar just follow these two steps: **Step 1)** Observe the path and retrace the steps that could have been taken to transpose one shape to the next. Can you logically say that it is a reflection, rotation, translation, or mix of them? If you can clearly see this, move on to step 2, otherwise they are not similar. **Step 2)** Verify that all the corresponding angles between the two shapes are congruent. If that holds up, make sure that the sides are either equivalent or proportional. If you can verify all those parameters, you have validated that the shapes are in fact similar.

Similarity transformations and self-similarity are crucial when learning the concepts of iterated function systems and fractals. At first this skill seems difficult but gets easier with experience. You will not only apply this concept to geometric shapes, but as we move on to advanced forms of statistical logic and working with matrix being able to distinguish analogous patterns will be very valuable.