# Transformations

Contents

In this topic you will learn about the most useful math concept for creating video game graphics: geometric transformations, specifically translations, rotations, reflections, and dilations.
You will learn how to perform the transformations, and how to map one figure into another using these transformations.

## Introduction to rigid transformations

The three rigid transformations are translations, rotations, and reflections. Get to know them and gain some experience in performing them on different figures.

7:23

Intro to geometric transformations

Sal introduces geometric transformations! Specifically, he explains what the "image" of a transformations is, what are the "rigid" transformations, and which transformations are not rigid.

Article

Intro to translations

Learn what translations are and how to perform them in our interactive widget.

1:37

Performing translations

Sal shows how to perform a translation on a triangle using our interactive widget!

Exercise

Perform translations

Use the interactive transformation tool to perform translations.

Article

Properties of translations

Learn and verify three important properties of geometrical translations.

Article

Intro to rotations

Learn what rotations are and how to perform them in our interactive widget.

1:25

Performing rotations

Sal shows how to perform a rotation on a pentagon using our interactive widget!

Exercise

Perform rotations

Use the interactive transformation tool to perform rotations.

1:48

Performing reflections

Sal shows how to perform a reflection on a quadrilateral using our interactive widget!

Exercise

Perform reflections

Use the interactive transformation tool to perform reflections.

## Translations

Learn how to find the translation that maps a given figure to another, and how to manually draw the image of a translation.

Article

Translating shapes

Learn how to draw the image of a given shape under a given translation.

1:55

Translating shapes

Sal is given a triangle on the coordinate plane and the definition of a translation, and he draws the image of the triangle under that translation.

Exercise

Translate shapes

Given a figure and a definition of a translation, manually draw the image.

Article

Determining translations

Learn how to find the necessary translation to map a given source shape onto a given image shape.

1:32

Determining translations

Sal is given two triangles on the coordinate plane, and determines the translation that maps one of them into the other.

Exercise

Determine translations

Given two figures on the coordinate plane, find the formal definition of the translation that takes one figure to another.

3:14

Translation challenge problem

A translation acting on the coordinate plane takes the point (-169,434) to point (-203, -68). What are the coordinates of the image of point (31, -529) under this translation?

## Rotations

Learn how to find the rotation that maps a given figure to another, and how to manually draw the image of a rotation.

Article

Determining rotations

Learn how to determine which rotation brings one given shape to another given shape.

2:46

Determining rotations

Sal is given two pairs of line segments and a center of a dilation. He finds the angle for the rotation that maps one pair of segments to another.

Exercise

Determine rotations

Given a figure on the coordinate plane and a center of a rotation, find the angle for the rotation that maps one figure to the other.

Article

Rotating shapes

Learn how to draw the image of a given shape under a given rotation about the origin by any multiple of 90°.

8:12

Rotating shapes

Sal is given a triangle on the coordinate plane and the definition of a rotation about the origin, and he manually draws the image of that rotation.

Exercise

Rotate shapes

Given a figure on the coordinate plane and the definition of a rotation about the origin, manually draw the image of that rotation.

10:32

Rotating shapes: center ≠ (0,0)

Sal is given a triangle on the coordinate plane and the definition of a rotation about an arbitrary point, and he manually draws the image of that rotation.

Exercise

Rotate shapes: center ≠ (0,0)

Given a figure on the coordinate plane and the definition of a rotation about an arbitrary point, manually draw the image of that rotation.

## Reflections

Learn how to find the reflection that maps a given figure to another, and how to manually draw the image of a reflection.

Article

Reflecting shapes

Learn how to find the image of a given reflection.

7:19

Reflecting shapes

Sal is given two line segments on the coordinate plane and the definition of a translation, and he draws the image of the segments under that reflection.

Exercise

Reflect shapes

Given a figure and a definition of a reflection, manually draw the image.

5:41

Determining reflections

Sal is given two line segments on the coordinate plane, and determines the reflection that maps one of them into the other.

Exercise

Determine reflections

Given two figures on the coordinate plane, find the formal definition of the reflection that takes one figure to another.

Exercise

Advanced reflections

Find the reflection that maps a given figure to another and draw the image of a reflection. The lines of reflection in this exercise have a wide range of slopes.

## Dilations or scaling around a point

In addition to the three rigid transformation, there are the dilations, which expand or shrink figures while keeping the same proportions. These are extremely important in the subject of Similarity!
Gain experience by performing dilation, learn how to find the dilation that maps a given figure to another, and learn how to manually draw the image of a dilation.

2:46

Performing dilations

Sal shows how to perform a dilation on a hexagon using our interactive widget!

Exercise

Perform dilations

Use the interactive transformation tool to perform dilations.

2:26

Dilating shapes: shrinking

Sal is given a triangle on the coordinate plane and he draws the image of the triangle under a dilation with scale factor 1/2 about the origin.

2:39

Dilating shapes: expanding

Sal is given a rectangle on the coordinate plane and he draws the image of the rectangle under a dilation with scale factor 1 2/3 about an arbitrary point.

Exercise

Dilate shapes

Given a figure and a definition of a dilation, manually draw the image.

3:59

Determining dilations

Sal shows how we can use dilations to map a line into another, parallel, line.

Exercise

Determine dilations

Find both the center and the scale factor of a dilation that maps a given figure to another one.

## Sequences of transformations

Learn how to analyze a sequence of different transformations performed in order.

Exercise

Sequences of transformations

Given a description of a sequence of transformations, determine whether it preserves segment length or angle measure.

## Properties and definitions of transformations

Let's continue our deep voyage through the world of transformations! Use your knowledge about the precise descriptions of the rigid transformations and their properties in various situations.

Article

Precisely defining rotations

Read a dialog where a student and a teacher work towards defining rotations as precisely as possible.

4:41

Identifying type of transformation

Sal is given information about a transformation in terms of a few pairs of points and their corresponding images, and he determines what kind of transformation it can be.

Exercise

Identify transformations

Given a description of the effect of a transformation, determine which rigid transformation it is.

## Symmetry

Learn about different kinds of symmetries of two-dimensional shapes, and analyze various shapes according to their symmetries.

2:47

Intro to reflective symmetry

Sal introduces the concept of an "axis of symmetry."

4:08

Intro to rotational symmetry

Sal checks whether various figure are symmetrical under a 180 degrees rotation.

3:07

Finding a quadrilateral from its symmetries

Two of the points that define a certain quadrilateral are (0,9) and (3,4). The quadrilateral has reflective symmetry over the line y=3-x. Draw and classify the quadrilateral.

7:20

Finding a quadrilateral from its symmetries (example 2)

Two of the points that define a certain quadrilateral are (-4,-2) and (0,5). The quadrilateral has a reflective symmetry over the lines y=x/2 and y=-2x+5. Draw and classify the quadrilateral.

Exercise

Symmetry of 2D shapes

Analyze various shapes according to their reflective and rotational symmetries.