1. Dilations
Lesson 6.7

2. What You Should Learn Why You Should Learn It
How to identify dilations
How to use dilations to solve real-life problems
You can use dilations to solve real-life problems, such as finding settings on photographic enlargers

3. Dilation Vocabulary
The center of dilation is a fixed point in the plane about which all points are expanded or contracted.  It is the only invariant point under a dilation.
Note: If the center of the pre-image is the origin, then the center of the image is also the origin.

4. Vocab Continued
Scale Factor- the factor that is used to produce the image from the pre-image
Pre-image- the starting figure
Image- the result of the pre-image multiplied by a scale factor

5. What is a dilation?
A type of nonrigid transformation where every image is similar to its preimage
A dilation with center C and scale factor k is a transformation that maps every point P in the plane to a point P ', so the following properties are true
1. If P is not point C , then CP ' is coincident to CP, and CP ' = k(CP), where k >0, and k1
2. If P is point C, the P = P '

6. Do Dilations Preserve?
Properties preserved (invariant) under a dilation: 1.  angle measures (remain the same) 2.  parallelism (parallel lines remain parallel) 3.  colinearity (points stay on the same lines) 4.  midpoint (midpoints remain the same in each figure) 5.  orientation (lettering order remains the same)
Not preserved?

7. Notation of scale factors (points)
k>0 (x,y) k(x,y) (kx,ky)
What happens when k<0?
(x,y) k(x,y) ((-k)x,(-k)y)
Thus the image is flipped across the line y=-x.

8. Example of a dilation
The dilation is a reduction if 0<k<1 and it is an enlargement if k > 1 5 2 1 2

9. Example 1
Identify the dilation and find its scale factor C

10. Example 1 Identify the dilation and find its scale factor
Reduction, Because
The scale factor is k = ⅔
Enlargement, Because
The scale factor is k = 2

11. Example 2 What is the scale factor of A(0,0), B(8, 16), C(4,0) To

12. Example 2 What is the scale factor of A(0,0), B(8, 16), C(4,0) To
Answer k = -1/4

13. Example 3 Dilation in a Coordinate Plane
Lesson 8.7
Example 3 Dilation in a Coordinate Plane
Draw a dilation of a rectangle ABCD with vertices A(1,1), B(3,1), C(3,2) and D(1,2). Use center as the origin. Use a scale factor of 2. How does the perimeter of the preimage compare to the perimeter of the image?

14. Example 3 Dilation in a Coordinate Plane
Draw a dilation of a rectangle ABCD with vertices A(1,1), B(3,1), C(3,2) and D(1,2). Use center as origin. Use a scale factor of 2.
How does the perimeter of the preimage compare to the perimeter of the image?
Preimage has a perimeter of 6
Image has a perimeter of 12
Note: The perimeter was enlarged by the same scale factor 2 as the points and sides have.

15. How about the area?
Will the area also change with the same scale factor as the perimeter did?

16. The answer is…
NO!!!!
The area of the image increases by a scale factor of n^2. Thus in our previous example the image has area 4 times larger than the pre-image.

17. THE END