Text of Dilations Lesson 6.7. What You Should Learn Why You Should Learn It How to identify dilations How to...
What You Should LearnWhy You Should Learn ItHow to identify dilationsHow to use dilations to solve real-life problemsYou can use dilations to solve real-life problems, such as finding settings on photographic enlargers
Dilation VocabularyThe 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.
Vocab ContinuedScale Factor- the factor that is used to produce the image from the pre-image Pre-image- the starting figureImage- the result of the pre-image multiplied by a scale factor
What is a dilation?A type of nonrigid transformation where every image is similar to its preimageA 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 true1. If P is not point C , then CP ' is coincident to CP, and CP ' = k(CP), where k >0, and k12. If P is point C, the P = P '
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?
Notation of scale factors (points)k>0 (x,y) k(x,y) (kx,ky)
What happens when k
Example of a dilationThe dilation is a reduction if 0
Example 1Identify the dilation and find its scale factorC
Example 1Identify the dilation and find its scale factorCReduction, BecauseThe scale factor is k = Enlargement, BecauseThe scale factor is k = 2
Example 2What is the scale factor of A(0,0), B(8, 16), C(4,0) ToA(0,0), B(-2,-4), C(-1,0)?
Example 2What is the scale factor of A(0,0), B(8, 16), C(4,0) To A(0,0), B(-2,-4), C(-1,0)?
Answer k = -1/4
Example 3Dilation in a Coordinate PlaneDraw 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?
Example 3Dilation in a Coordinate PlaneDraw 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 6Image has a perimeter of 12Note: The perimeter was enlarged by the same scale factor 2 as the points and sides have.
How about the area?Will the area also change with the same scale factor as the perimeter did?
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.