Lesson 2Circular Grid

Learning Goal

Let’s dilate figures on circular grids.

Learning Targets

I can apply dilations to figures on a circular grid when the center of dilation is the center of the grid.

Lesson Terms

center of dilation
dilation
scale factor

Warm Up: Notice and Wonder: Concentric Circles

What do you notice? What do you wonder?

Activity 1: A Droplet on the Surface

Problem 1

The larger circle $d$ is a dilation of the smaller circle $c$ . $P$ is the center of dilation.

Draw four points on the smaller circle using the Point on Object tool.
Draw the rays from $P$ through each of those four points. Select the Ray tool, then point $P$ , and then the second point.
Mark the intersection points of the rays and circle d by selecting the Intersect tool and clicking on the point of intersection.

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The larger Circle $d$ is a dilation of the smaller Circle $c$ . $P$ is the center of dilation.

Draw four points on the smaller circle (not inside the circle!), and label them $E$ , $F$ , $G$ , and $H$ .
Draw the rays from $P$ through each of those four points.
Label the points where the rays meet the larger circle $E^{'}$ , $F^{'}$ , $G^{'}$ , and $H^{'}$ .

A circular grid with center point P, and blue circle c and green circle d.

Problem 2

Complete the table. In the row labeled $c$ , write the distance between $P$ and the point on the smaller circle in grid units. In the row labeled $d$ , write the distance between $P$ and the corresponding point on the larger circle in grid units.

	$E$	$F$	$G$	$H$
$c$
$d$

Problem 3

The center of dilation is point $P$ . What is the scale factor that takes the smaller circle to the larger circle? Explain your reasoning.

Activity 2: Quadrilateral on a Circular Grid

Here is a polygon $A B C D$

- Dilate each vertex of polygon $A B C D$ using $P$ as the center of dilation and a scale factor of 2.
- Draw segments between the dilated points to create a new polygon.
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What are some things you notice about the new polygon?
Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?
Dilate each vertex of polygon $A B C D$ using $P$ as the center of dilation and a scale factor of $\frac{1}{2}$ .
What do you notice about this new polygon?

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Here is a polygon $A B C D$

Dilate each vertex of polygon $A B C D$ using $P$ as the center of dilation and a scale factor of 2. Label the image of $A$ as $A^{'}$ , and label the images of the remaining three vertices as $B^{'}$ , $C^{'}$ , and $D^{'}$ .
- Draw segments between the dilated points to create polygon $A^{'} B^{'} C^{'} D^{'}$ .
What are some things you notice about the new polygon?
Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?
Dilate each vertex of polygon $A B C D$ using $P$ as the center of dilation and a scale factor of $\frac{1}{2}$ . Label the image of $A$ as $E$ , the image of $B$ as $F$ , the image of $C$ as $G$ and the image of $D$ as $H$ .
What do you notice about polygon $E F G H$ ?

Are you ready for more?

Suppose $P$ is a point not on $\overset{―}{W X}$ . Let $\overset{―}{Y Z}$ be the dilation of $\overset{―}{W X}$ using $P$ as the center with scale factor 2. Experiment using a circular grid to make predictions about whether each of the following statements must be true, might be true, or must be false.

$\overset{―}{Y Z}$ is twice as long $\overset{―}{W X}$ .
$\overset{―}{Y Z}$ is five units longer than $\overset{―}{W X}$ .
The point $P$ is on $\overset{―}{Y Z}$ .
$\overset{―}{Y Z}$ and $\overset{―}{W X}$ intersect.

Activity 3: A Quadrilateral and Concentric Circles

Dilate polygon $E F G H$ using $Q$ as the center of dilation and a scale factor of $\frac{1}{3}$ . The image of $F$ is already shown on the diagram. (You may need to draw more rays from $Q$ in order to find the images of other points.)

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A circular grid with center point Q. A line is drawn from center Q with points F' and F. A figure EFGH intersects the line at F and surrounds center Q.

Lesson Summary

A circular grid like this one can be helpful for performing dilations.

The radius of the smallest circle is one unit, and the radius of each successive circle is one unit more than the previous one.

To perform a dilation, we need a center of dilation, a scale factor, and a point to dilate. In the picture, $P$ is the center of dilation. With a scale factor of 2, each point stays on the same ray from $P$ , but its distance from $P$ doubles:

A circular grid with center P. A blue triangle ABC surrounds center P. A larger triangle A'B'C" surrounds ABC.

Since the circles on the grid are the same distance apart, segment $P A^{'}$ has twice the length of segment $P A$ , and the same holds for the other points.