1. Geometry
4.5 Dilations

2. 4.5 Warm Up Day 1 Use the graph to find the indicated length.
Find the length of Find the length of
𝐷𝐸 and 𝐸𝐹 𝐵𝐶 and 𝐴𝐶
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3. 4.5 Warm Up Day 2
Plot the points in a coordinate plane. Then determine whether the quadrilaterals are congruent. 1. A(-3, 4), B(-3, 7), C(3, -4), D(3, -1) 2. E(7, -2), F(2, -2), G(3, -4), H(5, -4) 3. I(9, 2), J(0, 2), K(6, 9), L(6, 2) 4. M(7, -9), N(7, 0), P(8, -3), Q(-1, -3)
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4. 4.5 Essential Question What does it mean to dilate a figure?
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5. Goals Identify Dilations Make drawings using dilations.
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6. Rigid Transformations
Rotations
Translations
These were isometries:
The pre-image and the image were congruent.
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7. Dilation Dilations are non-rigid transformations.
The pre-image and image are similar, but not congruent.
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8. Dilation
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9. Dilation
Enlargement
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10. Dilation
Reduction
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11. Dilation R S C T
Center of Dilation
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12. Dilation R R S C T Center of Dilation 2CR CR CR February 28, 2019
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13. Dilation R R S S C T Center of Dilation 2CR CR 2CS CS CR CS
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14. Dilation R R S S C T Center of Dilation T 2CR CR 2CS CS CR CS CT CT
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15. Dilation RST ~ RST R R S S C T Center of Dilation T 2CR CR 2CS
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4.5 Dilations

16. Dilation Definition
A dilation with center C and scale factor k is a transformation that maps every point P to a point P’ so that the following properties are true:
If P is not the center point C, then the image point P’ lies on CP. The scale factor k is an integer such that
k  1 and
2. If P is the center point C, then P = P’.
3. The dilation is a reduction if 0 < |k| < 1, and an enlargement if |k| > 1.
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4.5 Dilations

17. Dilation Enlargement Scale Factor R R S S C T Center of Dilation T
2CR CR
2CS R S CR S CS CS C CT T
Center of Dilation CT
2CT T
Scale Factor
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18. Example 1 Reduction What type of dilation is this? F G F’ G’ C K’ H’ H
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19. Example 1 What is the scale factor? F 45 G Notice: F’ G’ 15 k < 1
Reduction 15 12 36
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20. Example 2
Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement
𝑘= 𝐶𝑃′ 𝐶𝑃
Notice:
k > 1
Enlargement
𝑘= 12 8
𝑘= 3 2
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21. Your Turn
Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement
𝑘= 𝐶𝑃′ 𝐶𝑃
Notice:
k < 1
Reduction
𝑘= 18 30
𝑘= 3 5
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22. Remember: The scale factor k is S. F.= Image Length Pre−image Length
If 0 < |k| < 1 it’s a reduction.
If |k| > 1 it’s an enlargement.
S. F.= Image Length Pre−image Length
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23. Coordinate Geometry Use the origin (0, 0) as the center of dilation.
The image of P(x, y) is P’(kx, ky).
Notation: P(x, y)  P’(kx, ky).
Read: “P maps to P prime”
You need graph paper, a ruler, pencil.
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24. Example 3 Graph ABC with A(1, 1), B(3, 6), C(5, 4). B C A
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25. Example 3 B’
Using a scale factor of k = 2, locate points A’, B’, and C’. P(x, y)  P’(kx, ky). C’
A(1, 1)  A’(2  1, 2  1) = A’(2, 2)
B(3, 6)  B’(2  3, 2  6) = B’(6, 12) B
C(5, 4)  C’(2  5, 2  4) = C’(10, 8) C A’ A
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26. Example 3 B’
Draw ABC. C’ B C A’ A
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27. Example 3 You’re done. B’ C’
Notice that rays drawn from the center of dilation (the origin) through every preimage point also passes through the image point. B C A’ A
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28. Example 4
Graph quad. KLMN with K(−4, 8), L(0, 8), M(4, 4), and N(−4, −4) and its image after a dilation with a scale factor of 3 4
x, y → x, y
K(−4, 8) →
K′(−3, 6)
L(0, 8) →
L′(0, 6)
M(4, 4) →
M′(3, 3)
N(−4, −4) →
N′(−3, −3)
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29. Your Turn Draw RSTV with R(0, 0) S(6, 3) T(0, 12) V(6, 3) T(0, 12)
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R(0, 0)

30. Your Turn Draw R’S’T’V’ using a scale factor of k = 1/3. T(0, 12)
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R(0, 0)
R’(0, 0)

31. Your Turn R’S’T’V’ is a reduction. T(0, 12) T’(0, 4) S(-6, 3) V(6, 3)
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R(0, 0)
R’(0, 0)

32. Negative Scale Factor
In the coordinate plane, you can have scale factors that are negative numbers.
When this occurs, the figure is dilated and rotates 180°.
The following is still true…
If 0 < |k| < 1 it’s a reduction.
If |k| > 1 it’s an enlargement.
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33. Example 5
Graph △FGH with vertices F(−4, −2), G(−2, 4), and H(−2, −2) and its image after a dilation with a scale factor of − 1 2
x, y → − 1 2 x, − y
F(−4, −2) →
G(−2, 4) →
H(−2, −2) →
F′(2, 1)
G′(1, −2)
H′(1, 1)
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34. Your Turn
Graph △PQR with vertices P(1, 2), Q(3, 1), and R(1, −3) and its image after a dilation with a scale factor of −2.
x, y → −2x, −2y
P(1, 2) →
Q(3, 1) →
R(1, −3) →
P’ (−2, −4)
Q’ (−6, −2)
R’ (−2, 6)
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35. Example 6
You are making your own photo stickers. Your photo is 4 inches by 4 inches. The image on the stickers is 1.1 inches by 1.1 inches. What is the scale factor of this dilation?
S. F.= Image Length Pre−image Length
S. F.=
S. F.=
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36. Your Turn
An optometrist dilates the pupils of a patient’s eyes to get a better look at the back of the eyes. A pupil dilates from 4.5 millimeters to 8 millimeters. What is the scale factor of this dilation?
S. F.= Image Length Pre−image Length
S. F.=
= 80 45
S. F.= 16 9
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37. Example 7
You are using a magnifying glass that shows the image of an object that is six times the object’s actual size. Determine the length of the image of the spider seen through the magnifying glass.
S. F.= Image Length Pre−image Length
6= x 1.5
x = 1.5 (6)
x = 9 cm
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38. Your Turn
You are using a magnifying glass that shows the image of an object that is six times the object’s actual size. The image of a spider is shown at the left. Find the actual length of the spider.
S. F.= Image Length Pre−image Length
6= x
6 1 = x
6x = 12.6
x = 2.1 cm
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39. Summary A dilation creates similar figures.
A dilation can be a reduction or an enlargement.
If the scale factor is less than one, it’s a reduction, and if the scale factor is greater than one it’s an enlargement.
A negative scale factor is the same as a dilation with a 180° rotation.
The scale factor is found using S. F.= Image Pre−image
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40. Assignment
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