# How to Dilate (given any center)

- Author:
- David Hornbeck

- Topic:
- Dilation

How do you determine the coordinates of an image to a dilation given any center and scale factor? In other words,

*how do you dilate?*Follow the activity below to investigate.Orient yourself with the file. Observe in the top left corner what s, c, & d are, as well as where the sliders are.
1. When s=2, what is the relationship between the coordinates of the image and the pre-image? Adjust the slider so that s=3,s=1,s= 1/2,s=0,s=-2,s=-3. Can you make a conjecture about the relationship between the scale factor s and the coordinates of the image?
2. Move the c and d sliders so that the center of dilation moves around. Does your conjecture from question 1 hold?
3. What changed from question 1 to question 2? Can you refine your conjecture from question 1?
4. Use the sliders to move the center of dilation D to (2, -1) and the scale factor to s=2. Right click anywhere in blank space until you see a menu with Graphics at the top. Click on Axes. Now, click on the “Show new coordinate plane” box. Describe what you see. What happened to the coordinate plane? What does the point O represent?
5. In question 5, you pretended that D was the origin; if it was, we could use the conjecture from question 3. However, it is not. What geometric transformation could you use to move the entire coordinate plane (along with D and the triangles) so that O is the origin again? Drag the coordinate plane accordingly.
6. Click the “Show first auxiliary triangle” box. (Note: “auxiliary” describes something that helps or supports.) What happened to ∆ABC? Does this agree with your answer to question 5?
7. Click the “Show second auxiliary triangle.” Describe specifically what happens.
8. What geometric transformation would transform the second auxiliary triangle into ∆A^' B^' C^'? Is there anything significant about this? (Ring any bells?)
9. Click the “Show third auxiliary triangle” box. Were you correct in 8?
10. Review your answers to the previous question and complete the following table to create a rule for dilating any point (x,y) from any center of dilation (c, d) by any scale factor s. (The table does not appear in this format, but it is easy to imagine where to fill things in.)
(HINT: In questions 4 – 8, c=2,d=-1,s=2. Three transformations occurred to get ∆ABC to ∆A'B'C'.)
Mathematical Representation Transformation Occurring In My Own Words
(What’s happening and why?)
(x,y) → ( , )
( , ) → ( , )
( , ) → ( , )
10. Click the third tool on the tool bar. Select Line through Two Points. Click on point A, then A’. Repeat for B & B’ and C & C’. Write down your observation about the three lines.
11. Use the sliders to change c, d, & s. Move the vertices A, B, and C around as much as you like. Can you make a conjecture related to the three lines? Be specific.