Lesson 3Dilations with no Grid

Learning Goal

Let’s dilate figures not on grids.

Learning Targets

I can apply a dilation to a polygon using a ruler.

Lesson Terms

dilation
scale factor

Warm Up: Points on a Ray

Find and label a point $C$ on the ray whose distance from $A$ is twice the distance from $B$ to $A$ .
Find and label a point $D$ on the ray whose distance from $A$ is half the distance from $B$ to $A$ .

A line segment with endpoint A and another point B.

Activity 1: Dilation Obstacle Course

open applet in presentation mode

Dilate $B$ using a scale factor of 5 and $A$ as the center of dilation. Which point is its image?
Using $H$ as the center of dilation, dilate $G$ so that its image is $E$ . What scale factor did you use?
Using $H$ as the center of dilation, dilate $E$ so that its image is $G$ . What scale factor did you use?
To dilate $F$ so that its image is $B$ , what point on the diagram can you use as a center?
Dilate $H$ using $A$ as the center and a scale factor of $\frac{1}{3}$ . Which point is its image?
Describe a dilation that uses a labeled point as its center and that would take $F$ to $H$ .
Using $B$ as the center of dilation, dilate $H$ so that its image is itself. What scale factor did you use?

Print Version

Here is a diagram that shows nine points.

Dilate $B$ using a scale factor of 5 and $A$ as the center of dilation. Which point is its image?
Using $H$ as the center of dilation, dilate $G$ so that its image is $E$ . What scale factor did you use?
Using $H$ as the center of dilation, dilate $E$ so that its image is $G$ . What scale factor did you use?
To dilate $F$ so that its image is $B$ , what point on the diagram can you use as a center?
Dilate $H$ using $A$ as the center and a scale factor of $\frac{1}{3}$ . Which point is its image?
Describe a dilation that uses a labeled point as its center and that would take $F$ to $H$ .
Using $B$ as the center of dilation, dilate $H$ so that its image is itself. What scale factor did you use?

Activity 2: Getting Perspective

Dilate $P$ using $C$ as the center and a scale factor of 4. Follow the directions to perform the dilations in the applet.
1. Select the Dilate From Point tool.
2. Click on the object to dilate, and then click on the center of dilation.
3. When the dialog box opens, enter the scale factor. Fractions can be written with plain text, ex. 1/2.
4. Click
5. Use the Ray tool and the Distance tool to verify.
Dilate $Q$ using $C$ as the center and a scale factor of $\frac{1}{2}$ .
open applet in presentation mode
Draw a simple polygon.
1. Choose a point outside the polygon to use as the center of dilation. Label it $C .$
2. Using your center $C$ and the scale factor you were given, draw the image under the dilation of each vertex of the polygon, one at a time. Connect the dilated vertices to create the dilated polygon.
3. Draw a segment that connects each of the original vertices with its image. This will make your diagram look like a cool three-dimensional drawing of a box! If there’s time, you can shade the sides of the box to make it look more realistic.
open applet in presentation mode
Compare your drawing to other people’s drawings. What is the same and what is different? How do the choices you made affect the final drawing? Was your dilated rectangle closer to $C$ than to the original rectangle, or farther away? How is that decided?

Print Version

Using one colored pencil, draw the images of points $P$ and $Q$ using $C$ as the center of dilation and a scale factor of 4. Label the new points $P^{'}$ and $Q^{'}$ .

Using a different color, draw the images of points $P$ and $Q$ using $C$ as the center of dilation and a scale factor of $\frac{1}{2}$ . Label the new points $P^{″}$ and $Q^{″}$ .
Pause here so your teacher can review your diagram. Your teacher will then give you a scale factor to use in the next part.
Now you’ll make a perspective drawing. Here is a rectangle.
- Choose a point inside the shaded circular region but outside the rectangle to use as the center of dilation. Label it $C$ .
- Using your center $C$ and the scale factor you were given, draw the image under the dilation of each vertex of the rectangle, one at a time. Connect the dilated vertices to create the dilated rectangle.
- Draw a segment that connects each of the original vertices with its image. This will make your diagram look like a cool three-dimensional drawing of a box! If there’s time, you can shade the sides of the box to make it look more realistic.
Compare your drawing to other people’s drawings. What is the same and what is different? How do the choices you made affect the final drawing? Was your dilated rectangle closer to $C$ than to the original rectangle, or farther away? How is that decided?

Are you ready for more?

Here is line segment $D E$ and its image $D^{'} E^{'}$ under a dilation.

Use a ruler to find and draw the center of dilation. Label it $F$ .
What is the scale factor of the dilation?

Lesson Summary

If $A$ is the center of dilation, how can we find which point is the dilation of $B$ with scale factor 2?

A ray with endpoint A and points D, B, and C moving away from A.

Since the scale factor is larger than 1, the point must be farther away from $A$ than $B$ is, which makes $C$ the point we are looking for. If we measure the distance between $A$ and $C$ , we would find that it is exactly twice the distance between $A$ and $B$ .

A dilation with scale factor less than 1 brings points closer. The point $D$ is the dilation of $B$ with center $A$ and scale factor $\frac{1}{3}$ .

Lesson 3Dilations with no Grid

Learning Goal

Learning Targets

Lesson Terms

Warm Up: Points on a Ray

Problem 1

Activity 1: Dilation Obstacle Course

Problem 1

Activity 2: Getting Perspective

Problem 1

Are you ready for more?

Problem 1

Lesson Summary