Geometry: Dilations

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

In a dilation

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 ____________________________

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

Why is ?

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)

Let’s consider why this theorem is true.

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 ?

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?

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 __________.