1. Geometry: Dilations

2. We have already discussed translations, reflections and rotations. Each of these transformations is an isometry, which means

3. In a dilation

4. A dilation centered at point C with a scale factor of k, where can be defined as follows: 1. The image of point C is itself. That is, _____ 2. For any point P other than C, the ____________________________

5. NOTE: If, then the dilation is a ____________ If, then the dilation is an ____________

6. Why is ?

8. Example: Under a dilation, triangle A(0,0), B(0,4), C(6,0) becomes triangle A'(0,0), B'(0,10), C'(15,0). What is the scale factor for this dilation? B’(0, 10) C’(15, 0)

9. Let’s consider why this theorem is true.

10. Example: Line segment AB with endpoints A(2, 5) and B(6, -1) lies in the coordinate plane. The segment will be dilated with a scale factor of and a center at the origin to create. What will be the length of ?

11. Example: Under a dilation of scale factor 3 with the center at the origin, what will be the coordinates of the image of point A(3, 4)? point B(4,1)? What do you notice about the coordinates of points A and A’ as well as B and B’ in relation to the scale factor?

12. Theorem: If the center of dilation is the origin and the scale factor is k, the coordinates of the point A’, the image of A(x, y), will be __________.