11.2 Proving Figures are Similar Using Transformations

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Name
Class
Date
11.2 Proving Figures are Similar
Using Transformations
Essential Question: How can similarity transformations be used to show two figures are
similar?
Resource
Locker
Confirming Similarity
Explore
A similarity transformation is a transformation in which an image has the same shape as its pre-image. Similarity
transformations include reflections, translations, rotations, and dilations. Two plane figures aresimilar if and only if
one figure can be mapped to the other through one or more similarity transformations.
A grid shows a map of the city park. Use tracing paper to confirm that the park elements are similar.
A
Trace patio EFHG. Turn the paper so that patio EFHG is mapped onto patio LMON.
Describe the transformation. What does this confirm about the patios?
y
F
6
4
E
H
2
-8
-6
-4N -2
2
G
4
x
6
8
-2
L
-4
O
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0
-6
M
A rotation of 180° around the origin; The fountains are similar.
B
Trace statues ABCDEF and JKLMNO. Fold the paper so that statue ABCDEF is mapped onto
statue JKLMNO. Describe the transformation. What does this confirm about the statues?
y
K
L
-6
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A
C
D
-4
N
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J 2
M
B
4
O
-2
0
-2
2
x
4
F
6
E
A reflection across the y-axis; The figures are
similar.
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DO NOT
Correcti
 Describe the transformation you can use to map vertices of garden
RST to corresponding
vertices of garden DEF. What does this confirm about the gardens?
4
y
S
2 E
x
-8
-6
-4
-2
D
R
0
2
-2
4
F
6
8
T
1
A dilation with scale factor _
; The
2
gardens are similar.
Reflect
1.
Look back at all the steps. Were any of the images congruent to the pre-images? If so, what types
of similarity transformations were performed with these figures? What does this tell you about the
relationship between similar and congruent figures?
The two figures in Steps A and B are congruent to each other. The types of transformations
that were performed with these figures were rotations and reflections, which are rigid
motions. So congruent figures are also similar figures.
2.
If two figures are similar, can you conclude that corresponding angles are congruent? Why or why not?
Yes, the corresponding angles are congruent because rigid motions and dilations preserve
angle measures.
Determining If Figures are Similar
Explain 1
You can represent dilations using the coordinate notation(x, y) → (kx, ky), where k is the scale factor and the center
of dilation is the origin. If 0 < k < 1, the dilation is a reduction. If k > 1, the dilation is an enlargement.
Example 1 Determine whether the two figures are similar using similarity

△RST and △XYZ
X
To map △RST onto △XYZ, there must be some factor k that dilates
△RST.
y
R
x
-5
0
T
Z
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5
S
Y
Pre-image
Image
R(0, 1)
X(0, 3)
S(1, -1)
T(-1, -1)
588
Y(3, -3)
Z(-3, -3)
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transformations. Explain.
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You can see that each coordinate of the pre-image is multiplied by 3 to get the image, so this is a
dilation with scale factor 3. Therefore, △RST can be mapped onto △XYZ by a dilation with center at
the origin, which is represented by the coordinate notation (x, y) → (3x, 3y). A dilation is a similarity
transformation, so △RST is similar to △XYZ.
B
PQRS and WXYZ
10
To map PQRS onto WXYZ, there must
be some factor k that enlarges PQRS.
y
X(5, 9)
Y(12, 9)
8
6
W(5, 5)
Pre-image
Image
P(2, 2)
W(5, 5)
R(6, 4)
Y(12, 9)
Z(12, 5)
4 Q(2, 4)
R(6, 4)
2 P(2, 2)
S(6, 2)
Q(2, 4)
S(6, 2)
x
0
2
4
6
8
10
12
X(5, 9)
Z(12, 5)
14
Find each distance: PQ = 2, QR = 4 , WX = 4 , and XY = 7
If kPQ = WX, then k = 2. However. 2QR = / ≠ XY.
No value of k can be determine that will map PQRS to WXYZ.
So, the figures are/are not similar.
Your Turn
Determine whether the two figures are similar using similarity transformations. Explain.
3.
LMNO and GHJK
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14
12
y
K(2, 12)
J(8, 12)
10
8
6
4
2
0
O(1, 6)
N(4, 6)
G(2, 4)
L(1, 2)
2
H(8, 4)
M(4, 2)
4
x
6
8
10
Yes, you can use a dilation with scale factor 2 and center at the origin to map LMNO onto GHJK.
The figures are similar because a dilation is a similarity transformation.
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4.
△JKL and △MNP
DO NOT
Correcti
CDEF and TUVF
5.
J y
y
C
K
-5
L
0
M
T
x
5
U 0
-5
N
P
D
No, the angles are different. △JKL and
△MNP are not similar figures.
Explain 2
F
x
V
5
E
Yes, there is a dilation centered at point F with
a scale factor of 2. CDEF is similar to TUVF.
Finding a Sequence of Similarity Transformations
In order for two figures to be similar, there has to be some sequence of similarity transformations that maps one
figure to the other. Sometimes there will be a single similarity transformation in the sequence. Sometimes you must
identify more than one transformation to describe a mapping.
Example 2 Find a sequence of similarity transformations that maps the first figure to
the second figure. Write the coordinate notation for each transformation.

ABDC to EFHG
y A
5
B
D
C
F x
E
-5
0
5
G
EFHG is_
smaller than ABDC, the scale
_ factor k of the dilation must be between 0 and 1.
The length of AB is 4 and the length of EF is 2; therefore, the scale factor is __12 . Write the new coordinates
after the dilation:
A(1, 6)
Original Coordinates
Coordinates after
1
dilation k = __
2
(2 )
1
A′ __,3
B(5, 6)
C(-2, 2)
D(2, 2)
(2 )
C′(-1, 1)
D′(1, 1)
5
B′ __,3
A translation right 2 units and down 3 units completes the mapping.
( )
Coordinates after dilation
1,3
A′ _
2
Coordinates after
translation (x + 2, y - 3)
5
E __
,0
2
( )
( )
5,3
B′ _
2
( )
9
F __
,0
2
C′(-1, 1)
D′(1, 1)
G(1, -2)
H(3, -2)
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Since
H
The coordinates after translation are the same as the coordinates of
EFGH, so you can map ABDC to
EFHG by the dilation (x, y) → __12 x, __12 y followed by a translation (x, y) → (x + 2, y - 3).
(
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B
△JKL to △PQR
y
R
8
6
4
-8
-6
-4
J
0
-2
Q
K
2
x
L
2
P4
6
8
You can map △JKL to △PQR with a reflection across the x-axis followed by a dilation followed by a
90 ° counterclockwise rotation about the origin.
°
Reflection: (x, y) → (x, -y) Dilation : (x, y) → ( 3x, 3y ) 90 counterclockwise rotation:
(x, y) → ( -y, x )
Reflect
6.
Using the figure in Example 3A, describe a single dilation that maps ABDC to EFHG.
By connecting the corresponding vertices, you can identify (4, -6) as the center of dilation
for a dilation with scale factor __12 that maps ABDC to EFHG.
7.
Using the figure in Example 3B, describe a different sequence of transformations that will map
△JKL to △PQR.
Reflect △JKL across the y-axis. Then dilate with center at the origin and scale factor 3. Then
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rotate 90° clockwise around the origin.
Your Turn
For each pair of similar figures, find a sequence of similarity transformations that
maps one figure to the other. Use coordinate notation to describe the transformations.
8.
PQRS to TUVW
y
Q
6
R
4
U
V
-8
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-6
-4
W
-2
2
T
0
Reflection: (x, y) → (-x, y)
S
P
Followed by…
x
2
4
6
You can map PQRS to TUVW by a reflection
followed by a dilation.
8
591
Dilation: (x, y) →
(__13 x, __31 y)
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9.
DO NOT
Correcti
△ABC to △DEF
y
F
8
You can map △ABC to △DEF by a rotation
about the origin 180° followed by a dilation
followed by a translation.
6
Rotation: (x, y) → (-x, -y)
Followed by…
4
2
D
E
x
-8
-6
B
-4
-2 A
0
2
4
6
8
(
)
3 _
Dilation: (x, y) → _
x, 3y
2 2
Followed by…
Translation: (x, y) (x - 3, y + 1.5)
-2
-4
C
10. Describe a sequence of similarity transformations that maps JKLMN to VWXYZ.
L
Translate JKLMN right 7 units so that J maps to V.
‹ ›
−
Reflect JKLMNacross JN .
M
K
1
Dilate JKLMN with center J and scale factor _
.
2
J
V
N
Z
W
X
Explain 3
Y
Proving All Circles Are Similar
You can use the definition of similarity to prove theorems about figures.
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Circle Similarity Theorem
All circles are similar.
Example 3 Prove the Circle Similarity Theorem.
Given: Circle C with center C and raduis r.
Circle D with center D and raduis s.
s
r
D
C
Prove: Circle C is similar to circle D.
To prove similarity, you must show that there is a sequence of similarity transformations that maps circle
to circle D.
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A
→
‾ .
Start by transforming circle C with a translation along the vector CD
s
D
r
C
r
⇀
CD
Circle C'
Through this
translation , the image of point C is
point D
.
Let the image of circle C be circle Cʹ. The center of circle Cʹ coincides with point
B
D
.
Transform circle Cʹ with the dilation with center of dilation D and scale factor _rs .
Circle Cʹ is made up of all the points at distance r
from point D .
_s
After the dilation, the image of circle Cʹ will consist of all the points at distance r × r = s
from point D.
translation followed by the dilation
These are the same points that form circle D . Therefore, the
translations and dilations are similarity transformations ,
maps circle C to circle D . Because
you can conclude that circle C is similar to circle D .
Reflect
11. Can you show that circle C and circle D are similar through another sequence of similarity
transformations? Explain.
_
Yes, can reflect circle C across the perpendicular bisector of CD, mapping point C to point D.
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Then, follow the same steps for dilation.
12. Discussion Is it possible that circle C and circle D are congruent? If so, does the proof of the similarity of
the circles still work? Explain.
Yes, if the ratio of each circle is the same positive value. The proof still works, because the
ratio of s to r is 1, so the dilation does not affect the size of the circle.
Elaborate
13. Translations, reflections, and rotations are rigid motions. What unique characteristic keeps dilations from
being considered a rigid motion?
Unlike the other transformations, dilations don't preserve distance, meaning the length
of the sides will not stay the same between the pre-image and its image. The lengths of
corresponding sides will be proportional according to the scale factor used.
14. Essential Question Check-In Two squares in the coordinate plane have horizontal and vertical sides.
Explain how they are similar using similarity transformations.
Possible answer: translate the bottom left vertex of one square to the other square. Then
dilate the first square by the ratio of the side lengths.
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