# Dilations

- Author:
- Shannon Rush

A dilation is a transformation in which a figure is enlarged or reduced with respect to a given point.
The point is called the "center of dilation." The scale factor is the ratio of the lengths of the corresponding sides of the image and the original.

## Summary

As you changed the scale factor of the dilation and the vertices of the triangle, what relationships did you observe between the points, segments and angle measures of the original and the imagine created through a dilation?

For a dilation with a scale factor of k, the rule for changing each point is

## Scale Factor Observations

The GeoGebra sketch below now as a table available to help you compare figures' side lengths.
In cell B1, type: =A'B'/AB
In cell B2, type: =B'C'/BC
In cell B3, type: =A'C'/AC
Then, use the arrow tool to change the scale factor and the vertices of the triangle. Observe the ratios of the corresponding side lengths.

## Observations about the effects of the scale factor

Return to the sketch above as needed for the following three questions: When the scale factor is greater than 1, how does the image compare to the original?

When the scale factor is 0<k<1, how does the image compare to the original?

When the scale factor is less than 0, how does the image compare to the original?

Now, explore the distance from the center of dilation to each point.
Using the above sketch, click in the graphics view with the triangles and your measurement tools should appear. Measure the distance from the center of dilation to each point and observe any relationships present.
You could also use the spreadsheet. Type the following in separate cells:
=DB
=DB'
=DB'/DB
Repeat the process for points C and C', and A and A'.
Think about how we can use this information to identify the center of dilation and the scale factor.

The center of dilation is point A(0,0). Find the scale factor used to create this image.

What is the scale factor used to dilate BCDE to B'C'D'E'?

## Can you do it on your own?

Use the GeoGebra tools to create a polygon and a point. Then dilate your created figure with a scale factor of 0.8 and 1.25.

## Challenge:

Determine what transformation(s) were used to create EFGH from ABCD. Be as specific as you would need to be so someone else could recreate this image.

What transformation rules were needed to create the above image?