Investigating Properties of Dilations
What is a Dilation?
"A dilation is a transformation that can change the size of a polygon but leaves the shape unchanged. A dilation has a center of dilation and a scale factor which together determine the position and size of the image of a figure after the dilation." (HMH Geometry)
Investigate the dilation below and then answer the questions.
- Change the scale factor. Be sure to look at values greater than 1 and values less than 1, as well as a scale factor of 1. You can even try some negative scale factors.
- Drag the center to different locations.
- Finally, drag some of the vertices of the original triangle that form one of the lines (A and B) to different locations. Try putting either in the same place as the center.
1) Move the center of dilation (point O) to the origin. Change the dilation number to any number between 0 and 3. What happens?
2) Move the center of dilation (point O) to point B. Change the dilation factor to any number between 0 and 3. What happens?
3) What does a scale factor of 1 do?
4) What does a scale factor greater than 1 do?
5) What does a scale factor less than 1 do?
Click on the "Show Lines" box. Move around the center of dilation and change the scale factor to see what happens.
6) What do you notice about the relationship between the two lines, AB and its image A'B'?
Click on the "Show Side Lengths and Areas" box. Position A at (1,8), B at (1,4), and C at (-2,4) (so that the side lengths are whole numbers).
7) What do you notice about the relationship between the side lengths of the original triangle and the corresponding side lengths of the image triangle?
You probably discovered some of these properties of dilations:
Properties of Dilations
- Dilations preserve angle measure.
- Dilations preserve betweenness.
- Dilations preserve collinearity.
- Dilations preserve orientation.
- Dilations map a line segment (the pre-image) to another line segment whose length is the product of the scale factor and the length of the pre-image.
- Dilations map a line not passing through the center of dilation to a parallel line and leave a line passing through the center unchanged.