# Lesson 3: Dilations with no Grid

Let’s dilate figures not on grids.

## 3.1: Points on a Ray

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

## 3.2: Dilation Obstacle Course

GeoGebra Applet nQh9SVzk

1. Dilate $B$ using a scale factor of 5 and $A$ as the center of dilation. Which point is its image?

2. Using $H$ as the center of dilation, dilate $G$ so that its image is $E$. What scale factor did you use?

3. Using $H$ as the center of dilation, dilate $E$ so that its image is $G$. What scale factor did you use?

4. To dilate $F$ so that its image is $B$, what point on the diagram can you use as a center?

5. Dilate $H$ using $A$ as the center and a scale factor of $\frac{1}{3}$. Which point is its image?

6. Describe a dilation that uses a labeled point as its center and that would take $F$ to $H$.

7. Using $B$ as the center of dilation, dilate $H$ so that its image is itself. What scale factor did you use?

## 3.3: Getting Perspective

Follow the directions to perform the dilations in the applet.

1. Dilate $P$ using $C$ as the center and a scale factor of 4.
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.
1. Dilate $Q$ using $C$ as the center and a scale factor of $\frac12$.

GeoGebra Applet QWj2ptWS

1. Draw a simple polygon. Choose a point not on the polygon to use as the center of dilation. Label it.
1. Using your center point and a scale factor your teacher gives you, 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.
2. Draw segments that connect 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.
3. 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 polygon closer to your center point than to the original, or farther away? How is that determined?

GeoGebra Applet dFYbT8He

## Summary

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

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 $\frac{1}{3}$.