Dilations
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.
Click the Dilate button below to show the dilation of the triangle with respect to the point. Use the slider to change the scale factor. Move the points of the original triangle. Essentially play with all the factors involved! As you do, pay attention to the effects on the points, segments, and angles.
Add observations to #1 in the google doc.
Scale Factor Observations
Looking at the points in the first GeoGebra sketch, how did the points change from the original to the image. In the google doc #2, record your guess for the rule that changes the points to the image.
Now, use the GeoGebra sketch below and the provided table 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 and record your answer in #3.
Scale Factor Observations: Answer the following questions in your google doc #4.
a) When the scale factor is greater than 1, how does the image compare to the original?
b) When the scale factor is 1, how does the image compare to the original?
c) When the scale factor is between 0 and 1, how does the image compare to the original?
d) When the scale factor is less than 0 (negative), 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. Record your idea in your google doc #5.
#6
The center of dilation is point A(0,0). Find the scale factor used to create this image and record your answer on your google doc #6.
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. Insert a screenshot of your work for #7.
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. Add your answer to #8 in the google doc.
What transformation rules were needed to create the above image?