Geometry: Dilations. We have already discussed translations, reflections and rotations. Each of...
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Transcript of Geometry: Dilations. We have already discussed translations, reflections and rotations. Each of...
Geometry:
Dilations
We have already discussed translations, reflections and
rotations. Each of these transformations is an isometry,
which means preimage. thetocongruent is image the
In a dilation
preimage. thesimilar to is image the
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 ____________________________
CC '
CPkCP' and CPon is P'point CPk
NOTE: If , then the dilation is a ____________
If , then the dilation is an
____________
10 k
1k
ncontractio
expansion
Why is ?''~ QCPCPQ
?similarityfor // SSSSASAA
reflexiveby CC
CPk
CQk CQk
CQ
CPk
CP
kk
11
SAS
?
''//
Why
QPPQ
Converse CAP
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)
ACk AC' ,definition By the
615 k
6
15k
Let’s consider why this theorem is true.
ABkBA ''
''CA'
CA then,
SASby B'CA'~CAB Since
BA
AB
CAk ''
1
BA
AB
k
ABkBA ''
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 ?
52
''BA''BA
135
4132
5
2'' BA
132526)4(1562 2222 AB
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 ofpoints A and A’ as well as B and B’ in relation to the scale factor?
)12,9('A )3,12('B
preimage. in the coordinate
ingcorrespond the times3
is image in the coordinateEach
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 __________.),( ykxk