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DilationsDilations
Lesson 6.7Lesson 6.7
What You Should LearnWhat You Should LearnWhy You Should Learn ItWhy You Should Learn It
How to identify dilationsHow to identify dilationsHow to use dilations to solve real-life How to use dilations to solve real-life
problemsproblemsYou can use dilations to solve real-You can use dilations to solve real-
life problems, such as finding life problems, such as finding settings on photographic enlargers settings on photographic enlargers
Dilation VocabularyDilation Vocabulary
The The center of dilation center of dilation is a fixed is a fixed point in the plane about which all point in the plane about which all points are expanded or contracted. points are expanded or contracted. It is the only invariant point under a It is the only invariant point under a dilation.dilation.
Note: If the center of the pre-image Note: If the center of the pre-image is the origin, then the center of the is the origin, then the center of the image is also the origin.image is also the origin.
Vocab ContinuedVocab Continued
Scale Factor- Scale Factor- the factor that is used the factor that is used to produce the image from the pre-to produce the image from the pre-image image
Pre-image-Pre-image- the starting figure the starting figureImage-Image- the result of the pre-image the result of the pre-image
multiplied by a scale factormultiplied by a scale factor
What is a dilation?What is a dilation?
A type of nonrigid transformation where A type of nonrigid transformation where every image is similar to its preimageevery image is similar to its preimage
A dilation with center A dilation with center CC and and scale scale factor factor kk is a transformation that maps is a transformation that maps every point every point PP in the plane to a point in the plane to a point P P ', ', so the following properties are trueso the following properties are true1. If 1. If PP is not point is not point C C , then , then CP CP ' is coincident ' is coincident
to to CPCP, and , and CPCP ' = ' = kk((CPCP), where ), where k k >0, and >0, and kk112. If 2. If PP is point is point CC, the , the PP = = PP ' '
Do Dilations Preserve?Do Dilations Preserve?
Properties preserved (invariant) under a Properties preserved (invariant) under a dilation:dilation:1. angle measures (remain the same)1. angle measures (remain the same)2. parallelism (parallel lines remain parallel)2. parallelism (parallel lines remain parallel)3. colinearity (points stay on the same lines)3. colinearity (points stay on the same lines)4. midpoint (midpoints remain the same in 4. midpoint (midpoints remain the same in each figure)each figure)5. orientation (lettering order remains the 5. orientation (lettering order remains the same)same)
Not preserved? Not preserved?
Notation of scale factors Notation of scale factors (points)(points)
k>0 (x,y) k(x,y) k>0 (x,y) k(x,y) (kx,ky)(kx,ky)
What happens when k<0?What happens when k<0?(x,y) -k(x,y) ((-k)x,(-k)y)(x,y) -k(x,y) ((-k)x,(-k)y)Thus the image is flipped across the Thus the image is flipped across the
line y=-x.line y=-x.
Example of a dilationExample of a dilation
The dilation is a The dilation is a reductionreduction if 0<k<1 and it if 0<k<1 and it is an is an enlargement enlargement if k > 1if k > 1
1
2
2
5
Example 1Example 1
Identify the dilation and find its scale Identify the dilation and find its scale factorfactor
C
Example 1Example 1
Identify the dilation and find its scale factorIdentify the dilation and find its scale factor
C
Reduction, Because
The scale factor is k = ⅔
' 2
3
CP
CP ' 2
1
CP
CPEnlargement, Because
The scale factor is k = 2
Example 2Example 2
What is the scale factor of What is the scale factor of A(0,0), B(8, 16), C(4,0) A(0,0), B(8, 16), C(4,0) ToToA’(0,0), B’(-2,-4), C’(-1,0)?A’(0,0), B’(-2,-4), C’(-1,0)?
Example 2Example 2
What is the scale factor of What is the scale factor of A(0,0), B(8, 16), C(4,0) A(0,0), B(8, 16), C(4,0) To To A’(0,0), B’(-2,-4), C’(-1,0)?A’(0,0), B’(-2,-4), C’(-1,0)?
Answer k = -1/4Answer k = -1/4
Example 3Example 3Dilation in a Coordinate PlaneDilation in a Coordinate Plane
Draw a dilation of a rectangle ABCD with vertices 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 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 origin. Use a scale factor of 2. How does the perimeter of the preimage compare to the perimeter perimeter of the preimage compare to the perimeter of the image?of the image?
Example 3Example 3Dilation in a Coordinate PlaneDilation in a Coordinate Plane
Draw a dilation of a rectangle Draw a dilation of a rectangle ABCD with vertices A(1,1), B(3,1), ABCD with vertices A(1,1), B(3,1), C(3,2) and D(1,2). Use center as C(3,2) and D(1,2). Use center as origin. Use a scale factor of 2. origin. Use a scale factor of 2.
How does the perimeter of the How does the perimeter of the preimage compare to the perimeter preimage compare to the perimeter of the image?of the image?
Preimage has a perimeter of 6Preimage has a perimeter of 6 Image has a perimeter of 12Image has a perimeter of 12 NoteNote: The perimeter was enlarged : The perimeter was enlarged
by the same scale factor 2 as the by the same scale factor 2 as the points and sides have.points and sides have.
How about the area?How about the area?
Will the area also Will the area also change with the change with the same scale factor same scale factor as the perimeter as the perimeter did?did?
The answer is…The answer is…
NO!!!!NO!!!!
The area of the image increases by a scale factor of n^2. The area of the image increases by a scale factor of n^2. Thus in our previous example the image has area 4 times Thus in our previous example the image has area 4 times larger than the pre-image. larger than the pre-image.