DS - Transformations and Similarity Lesson

Transformations and Similarity

You have learned in the past about 3 different transformations:

• Translation (Slide): A transformation that "slides" each point of a figure the same distance in the same direction
• Rotation: A transformation that turns a figure about a fixed point through a given angle and a given direction
• Reflection: A transformation that "flips" a figure over a line of reflection

As you look at each of these transformations think about whether or not each of these transformations changes the shape or size of the figure.

Transformation	Description	Picture
Reflection	In a reflection, a shape is "flipped" over a given line.	A figure is reflected over line j.
Rotation	A rotation rotates or "turns" a shape around a given point.	The figure is rotated around point A.
Translation	A translation will "slide" a shape from one location to another without turning it.	The figure is translated down and to the right.

You will notice that after each of these transformations the shape may have changed position or orientation, but has not changed shape or size. Because translations, rotations, and reflections do not change the size or shape of the figure, they are called isometries. Isometries are transformations that produce congruent figures. They preserve shape and size.

Transformation	Description	Picture
Dilation	In a dilation, the figure is increased or decreased in size. All dilations have a center and scale factor.	The figure is dilated with the center j (the red dot) with scale factor 2. For example, if one of the side lengths of the smaller figure is 3 cm, the corresponding side length in the larger figure would be 6 cm.

A dilation does change the size of a figure. Because of this, a dilation does not produce congruent figures. It produces similar figures. Similar figures are figures that have the same shape, but not the same size.

Imagine you are standing in front of a screen making shadow puppets. You put your hands up into the light and make shadows on the screen. If you move your hands further from the screen and closer to the light, what will happen to the shadow? Will it get bigger or smaller? It turns out, the further you are from the screen, the larger the shadow gets. It is this same idea that allows movie theaters to project very large pictures from very small film. The projector in your classroom is also an example of this. This change in size (but not shape) is called a dilation. We often think of dilations as an increase in size, but a dilation can make figures smaller.

Each dilation has a scale factor or ratio of dilation. This is a ratio that describes the change in size. They also have a pre-image (the original) and an image (copy). In order to describe the dilation, we also have to describe the center of dilation. This is the point from which the dilation is drawn.

Check out the videos below that demonstrate how these two determine a dilation.

Dilations on a Coordinate Plane

Dilations Not on a Coordinate Plane

In this video you will see how to find the scale factor and center of dilation if all you have is the image and pre-image.

Example

If the distance from A to C' is 6 and the distance from A to C is 2.

What is the scale factor? 

Solution: 

Scale factor = Distance to image / Distance to Pre-image

Scale Factor = 6/2 = 3

Remember to simplify the proportion.

Example

If the distance from B to B' is 8 and the distance from A to B is 2. What is the scale factor?

Solution:

Scale factor = distance of image to center/ distance of pre-image to center

Scale factor =(AB + BB')/AB = (2+8)/2

=10/2

=5

Example

If the distance from D to D' is 9 and the scale factor is 4. What is the distance from A to D. 

Solution:

Given distance DD' = 9

Scale factor = distance of image to center/ distance of pre-image to center

Scale factor = (AD + DD')/AD

Substitute Scale factor = 4 and DD' =9 in the equation 4 = (AD +9)/AD

Multiply AD on both sides:  4AD = AD + 9

Subtract AD both sides: 4AD- AD = 9

Simplify:  3AD = 9

Divide three on both sides:  AD = 3

Example

What is the length of segment AD? 

Solution:

AC'/AC = AD'/AD

12/3 = (AD+3)/AD

4 = (AD + 3)/AD 

Multiply AD both sides:  4AD = AD + 3

Subtract AD both sides: 3AD = 3

Divide 3 both sides: AD = 1

IMAGES CREATED BY GAVS