Math high School
congruence and
rigid Motion
exerciSeS
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Copyright © 2014 Pearson Education, Inc.
2
High School: Congruence and Rigid Motion
CONTENTS
ExErCiSES
ExErcisEs
LEssON 1: rigid MOtiON �������������������������������������������������������������������������� 5
LEssON 2: rEfLEctiONs ����������������������������������������������������������������������������� 6
LEssON 3: rOtatiONs ������������������������������������������������������������������������������ 10
LEssON 4: traNsLatiONs ��������������������������������������������������������������������� 14
LEssON 5: BEiNg PrEcisE ������������������������������������������������������������������������ 17
LEssON 6: cOMPOsitE traNsfOrMatiONs ��������������������������������� 21
LEssON 7: PuttiNg it tOgEthEr ����������������������������������������������������� 25
LEssON 11: dEfiNiNg cONgruENcE ������������������������������������������������� 26
LEssON 12: triaNgLE cONgruENcE critEria ���������������������������� 29
LEssON 13: usiNg cONgruENcE critEria ����������������������������������� 32
LEssON 14: sYMMEtriEs Of POLYgONs ��������������������������������������������� 35
aNswErs
LEssON 2: rEfLEctiONs �������������������������������������������������������������������������� 40
LEssON 3: rOtatiONs ������������������������������������������������������������������������������ 43
LEssON 4: traNsLatiONs ��������������������������������������������������������������������� 46
LEssON 5: BEiNg PrEcisE ������������������������������������������������������������������������ 49
LEssON 6: cOMPOsitE traNsfOrMatiONs ��������������������������������� 55
LEssON 11: dEfiNiNg cONgruENcE ������������������������������������������������� 58
Copyright © 2014 Pearson Education, Inc.
3
High School: Congruence and Rigid Motion
CONTENTS
ExErCiSES
aNswErs
LEssON 12: triaNgLE cONgruENcE critEria ���������������������������� 61
LEssON 13: usiNg cONgruENcE critEria ����������������������������������� 63
LEssON 14: sYMMEtriEs Of POLYgONs ��������������������������������������������� 65
Note: Some of these problems are designed to be delivered electronically.
Copyright © 2014 Pearson Education, Inc.
4
High School: Congruence and Rigid Motion
LEssON 1: rigid MOtiON
ExErcisEs
t
Review your end of unit assessment from the previous unit.
t
Write your wonderings about geometric transformations.
t
Write a goal stating what you plan to accomplish in this unit.
t
Based on your previous work, write three things you will do differently during this
unit to increase your success.
Copyright © 2014 Pearson Education, Inc.
5
High School: Congruence and Rigid Motion
LEssON 2: rEfLEctiONs
ExErcisEs
ExErcisEs
1.
Copyright © 2014 Pearson Education, Inc.
Draw the reflections of the following images across the dotted line.
6
High School: Congruence and Rigid Motion
LEssON 2: rEfLEctiONs
ExErcisEs
2.
Come up with several words that are symmetric (e.g. MOM). Describe the line of
symmetry for the words you pick.
3.
What kind of triangle would you get if you reflected the green line across the black
line and then connected their endpoints?
Copyright © 2014 Pearson Education, Inc.
7
High School: Congruence and Rigid Motion
LEssON 2: rEfLEctiONs
4.
ExErcisEs
Which of the following graphs shows the proper reflection of the figure across the line
segment AB?
y
5
B
4
3
2
1
–4
–3
–2
–1
1
2
3
4
x
5
–1
–2
–3
–4
A
–5
A
y
–3
–2
5
B
4
–4
y
B
5
3
3
2
2
1
1
–1
1
2
4
3
5
x
–4
–3
–2
–1
–1
–2
–2
–3
–3
D
5
A
–2
x
3
2
2
1
1
1
2
4
5
x
B
4
4
3
5
x
–4
–3
–2
–1
1
–1
–1
–2
–2
–3
–3
–4
–4
A
–5
Copyright © 2014 Pearson Education, Inc.
5
y
3
–1
4
5
B
4
–3
3
–5
y
–4
2
–4
A
–5
C
1
–1
–4
A
B
4
–5
8
2
3
High School: Congruence and Rigid Motion
LEssON 2: rEfLEctiONs
5.
ExErcisEs
Draw in the line of symmetry for the following reflection:
y
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 x
–6 –5 –4 –3 –2 –1
–2
–3
–4
6.
Draw the shape given by these coordinates, (5, 2) (4, –1) (1, 6), and then reflect that
shape across the line x = 0.
challenge Problem
7.
If you reflect all negative values (all values for which y < 0) of the function drawn
below across the x-axis, what will the result look like?
y
10
9
8
7
6
5
4
3
2
1
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1
–2
–3
–4
–5
–6
–7
–8
–9
–10
Copyright © 2014 Pearson Education, Inc.
1 2 3 4 5 6 7 8 9 10 x
9
High School: Congruence and Rigid Motion
LEssON 3: rOtatiONs
ExErcisEs
ExErcisEs
1.
Rotate the following figure 90º around the origin.
y
6
5
4
3
2
1
1 2 3 4 5 6 7 x
–7 –6 –5 –4 –3 –2 –1
–2
–3
–4
–5
–6
2.
Reflecting the object below across the x-axis and rotating the object 180º about the
point (2, 0) will result in images that are:
y
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1
–2
–3
–4
–5
–6
1 2 3 4 5 6 7 x
A Identical
B Rotations
C Reflections
D Translations
Copyright © 2014 Pearson Education, Inc.
10
High School: Congruence and Rigid Motion
LEssON 3: rOtatiONs
3.
ExErcisEs
This object is rotated clockwise 90º and then reflected across the y-axis. How far will
point A be from its original location (5, 5)?
y
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1
–2
–3
–4
–5
–6
–7
Copyright © 2014 Pearson Education, Inc.
A (5, 5)
1 2 3 4 5 6 7 x
11
High School: Congruence and Rigid Motion
LEssON 3: rOtatiONs
4.
ExErcisEs
Which of the following graphs shows the correct image of the initial graph rotated
120º counter clockwise about the origin?
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
4
6
8
10 x
–2
–4
–6
–8
–10
A
–10
–8
–6
–4
y
y
10
10
B
8
8
6
6
4
4
2
2
–2
2
4
6
8
–10
10 x
–8
–6
–4
–2
y
y
10
10
D
8
–4
–4
6
4
4
2
2
2
4
6
8
–2
Copyright © 2014 Pearson Education, Inc.
2
4
6
8
10 x
8
6
–2
10 x
–10
–10
–6
6
8
–8
–8
–88
6
–6
–
–6
–10
4
–4
–
–4
C
2
–2
–2
10 x
–10
–8
–6
–4
–2
–2
–4
–4
–6
–6
–8
–8
–10
–10
12
High School: Congruence and Rigid Motion
LEssON 3: rOtatiONs
5.
ExErcisEs
Construct the figure with the given coordinates, (3, 4) (6, 2) (5, 4) and then rotate the
figure 270º around the origin.
challenge Problem
6.
Draw a quadrilateral in the grid below and label the vertices. Draw its mapping after
it is rotated 90º clockwise around the point (3, –2) and then reflected across the x-axis.
y
10
8
6
4
2
–12 –10 –8
–6
–4
–2
–2
–4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
13
2
4
6
8
10
12 x
High School: Congruence and Rigid Motion
LEssON 4: traNsLatiONs
ExErcisEs
ExErcisEs
1.
A translation involves every x-value traveling a distance of h and every y-value
traveling three times the distance that the x-values travel. Which translation represents
this correctly?
A T(x, y) = (x + h, 3y)
B T(x, y) = (3x, y + h)
C T(x, y) = (x + 3h, y + h)
D T(x, y) = (x + h, y + 3h)
2.
Describe a translation that will shift the figure ABCD entirely into quadrant IV.
y
10
8
6
A
4
D
B
–10
–8
2
–6
–4
C
–2
2
–2
–4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
14
4
6
8
10 x
High School: Congruence and Rigid Motion
LEssON 4: traNsLatiONs
3.
ExErcisEs
Translate the figure EFG using the following translation: T(x, y) = (x – 4, y + 3).
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
–2
E
4
6
10 x
8
G
–4
–6
F
–8
–10
4.
Describe the translation that is shown:
y
10
A' 8
6
A
4
D'
C'
2
B
–10
–8
–6
B'
–4
–2
2
–2
D
C
–4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
15
4
6
8
10 x
High School: Congruence and Rigid Motion
LEssON 4: traNsLatiONs
5.
ExErcisEs
Translate the following figure 2 units in the positive x-direction, and 3 units in the
negative y-direction.
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
4
6
8
10 x
–2
–4
–6
–8
–10
challenge Problem
6.
Copyright © 2014 Pearson Education, Inc.
Create your own figure with at least 4 points, and then translate it with the following
translation T(x, y) = (x – 8, y + 4).
16
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
ExErcisEs
ExErcisEs
1.
Perform a vertical stretch on the following figure, using the stretch that takes the
y-coordinate of each point and doubles it. (Note that the x-coordinate of each
point is unchanged.)
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
2
4
6
10 x
8
–2
2
–4
–6
–8
–10
2.
Perform a horizontal stretch on the following figure. Use the stretch that takes the
x-coordinate of each point and triples it. (Note that the y-coordinate of each
point is unchanged.)
y
10
8
6
4
2
–14
–12
–10
–8
–6
–4
–2
2
–2
–4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
17
4
6
8
10
12
14
x
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
3.
ExErcisEs
Perform a dilation on the following figure, with the origin as the center point, and a
scale factor of 4.
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
4
6
8
10 x
–2
–4
–6
–8
–10
4.
Copyright © 2014 Pearson Education, Inc.
Create a figure with at least 4 points, and demonstrate a rigid motion transformation
and a non-rigid transformation on the same figure. Explain the different effects
you notice.
18
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
5.
ExErcisEs
Perform a horizontal stretch and a vertical translation on the following figure.
Show an image for each transformation separately, and then one image with both
transformations done.
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
2
–2
2
–4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
19
4
6
8
10 x
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
ExErcisEs
challenge Problem
6.
Perform the following transformation T(x, y) = (3x + 2, 4y – 3) on the given figure, then
describe what types of motion this transformation describes.
y
12
10
8
6
4
2
–12 –10
–8
–6
–4
–22
2
–2
2
–4
–6
–8
–10
–12
Copyright © 2014 Pearson Education, Inc.
20
4
6
8
10
12
x
High School: Congruence and Rigid Motion
LEssON 6: cOMPOsitE traNsfOrMatiONs
ExErcisEs
ExErcisEs
1.
Perform the following transformations on the figure ABCD, then graph the final image.
T(3, 4)
R(0, 0), 45°
y
10
8
6
4
2
C
–10
–8
–6
–4
–2
2
–2
4
6
B
8
10
12 x
D
A
–4
–6
–8
–10
2.
Perform a series of two transformations that moves the figure EFG completely into
quadrant II.
y
10
8
6
4
2
–10
–8
–6
–4
–2
–2
–4
F
2
4
G
–66
E
–8
–10
Copyright © 2014 Pearson Education, Inc.
21
6
8
10 x
High School: Congruence and Rigid Motion
LEssON 6: cOMPOsitE traNsfOrMatiONs
3.
ExErcisEs
The figure ABC has been translated 6 units to the right (A'B'C') and then rotated
(A"B"C"), find another sequence of transformations that will result in the same final
image.
y
10
8
B"
6
A"
4
C"
–10
–8
–6
–4
B
2
–2
B'
–2
–44
–6
A
2
4
6
C
8
10 x
C'
A'
–8
–10
4.
Determine the sequence of transformations with the fewest steps required to move the
figure JKL to the image J'K'L' shown.
y
10
8
6
J
L4
K'
2
L'
K
–10
–8
–6
–4
–2
2
4
–2
–4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
22
6
8
10 x
J'
High School: Congruence and Rigid Motion
LEssON 6: cOMPOsitE traNsfOrMatiONs
5.
ExErcisEs
The figure ABC has been rotated 45º, then 30º, and then 95º around the origin.
What one compound rotation will result in the same final image?
y
10
8
C"'
6
4
A"'
2
B"'
–10
–8
–6
–4
–2
2
–2
4
6
B
8
10 x
A
–4
–6
C
–8
–10
6.
Figure DEF has been translated 4 units to the left, then 3 units down, then 5 units to the
right, then 6 units up. What one compound translation will result in the same
final image?
y
10
8
6
D""
4
D
–10
–8
–6
6
E""
–4
4 E –2
2
2
F
F""
2
4
–4
4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
23
6
8
10 x
High School: Congruence and Rigid Motion
LEssON 6: cOMPOsitE traNsfOrMatiONs
ExErcisEs
challenge Problem
7.
Perform the following transformations on the figure GHIJ, and graph the final image.
T(7, 0)
R(0, 0), –90°
r y=0
y
10
8
6
G
H
4
2
–10
–8
–66
J
–4
4
I
–2
2
4
6
8
10 x
–2
–4
–6
–8
–10
Explain whether or not the order in which you do these transformations matters for
determining the final image.
Copyright © 2014 Pearson Education, Inc.
24
High School: Congruence and Rigid Motion
LEssON 7: PuttiNg it tOgEthEr
ExErcisEs
t
Read through your Self Check and think about your work in this lesson.
t
Write down what you have learned during the lesson.
t
What would you do differently if you were starting Self Check task now?
t
Which method would you prefer to use if you were doing the task again? Why?
t
Compare the new approaches you learned about with your original method.
t
Record your ideas— keep track of problem-solving strategies.
t
Complete any exercises from this unit you have not finished.
Copyright © 2014 Pearson Education, Inc.
25
High School: Congruence and Rigid Motion
LEssON 11: dEfiNiNg cONgruENcE
ExErcisEs
ExErcisEs
1.
Determine if these two triangles are congruent by showing a sequence of rigid
motions that maps ABC to DEF.
y
8
E
6
F
B
4
D
2
C
A
–2
2
4
6
10 x
8
–2
2.
Determine if these two triangles are congruent. Describe your method and justify
your response.
y
6
E
D
–6
F
2
–4
–2
A
2
6 x
4
–2
B
Copyright © 2014 Pearson Education, Inc.
4
–4
C
–6
26
High School: Congruence and Rigid Motion
LEssON 11: dEfiNiNg cONgruENcE
3.
ExErcisEs
Determine if triangle JKL and triangle MNO are congruent. Describe your method and
justify your response.
y
4
J
2
–8
–6
K
–4
L
–2
2
–2
4
6
8
x
N
–4
O
–6
M
–8
4.
Show that the triangle ABC and triangle DBC are congruent. Describe your method
and justify your response.
6
B
6
A
65°
65°
85°
10
D
85°
10
30°
30°
C
Copyright © 2014 Pearson Education, Inc.
27
High School: Congruence and Rigid Motion
LEssON 11: dEfiNiNg cONgruENcE
5.
ExErcisEs
Show that figures DEF and GHI are congruent by showing each pair of sides and each
pair of angles are congruent.
y
F
6
4
D
2
E
–6
–4
I
–2
2
6 x
4
–2
G
–4
H
–6
6.
Determine whether or not triangles JKL and MNO are congruent by showing if each
pair of sides and each pair of angles is congruent. If they are not congruent, show how
you know.
y
10
O
8
L
6
4
N
2
J
–10
–8
K
–6
–4
M
–2
2
4
6
8
10 x
–2
challenge Problem
7.
Copyright © 2014 Pearson Education, Inc.
Create a triangle that lies completely in quadrant III, and then create a congruent
triangle that is rotated from the original and lies completely within quadrant I.
28
High School: Congruence and Rigid Motion
LEssON 12: triaNgLE cONgruENcE critEria
ExErcisEs
ExErcisEs
For problems 1–5, determine whether or not each pair of triangles is congruent.
Explain which congruence criteria you used for each case.
A
1.
B
5 cm
90°
C
90°
5 cm
D
2.
A
5 cm
B
C
D
8 cm
E
3.
C
22°
A
40°
D
40° 22°
B
Copyright © 2014 Pearson Education, Inc.
29
High School: Congruence and Rigid Motion
LEssON 12: triaNgLE cONgruENcE critEria
4.
ExErcisEs
A
15 cm
120°
D
B
15 cm
F
120°
12 cm
E
5.
C
12 cm
3 cm
4 cm
5 cm
4 cm
5 cm
3 cm
6.
Which missing piece (angle or side measurement) do you need in order to be sure that
these two triangles are congruent?
D
A
5 cm
5 cm
B
Copyright © 2014 Pearson Education, Inc.
E
4 cm
4 cm
F
C
30
High School: Congruence and Rigid Motion
LEssON 12: triaNgLE cONgruENcE critEria
ExErcisEs
challenge Problem
7.
Prove that trapezoids ABCD and EFGH are congruent, either by showing a rigid motion
that links the two figures, or by showing that every matching pair of sides and angles
is congruent.
y
8
6
4
A
E
B
F
C
G
2
–2
2
–2
Copyright © 2014 Pearson Education, Inc.
D
4
6
H
10 x
8
31
High School: Congruence and Rigid Motion
LEssON 13: usiNg cONgruENcE critEria
ExErcisEs
ExErcisEs
For problems 1–6, determine whether or not the two triangles shown are congruent.
Explain the congruence criteria that you used for each figure.
1.
AD is parallel to BC, and AB is parallel to DC:
B
A
C
D
2.
C is the center of the green circle. BE is a diameter of the circle.
B
A
C
D
Copyright © 2014 Pearson Education, Inc.
E
32
High School: Congruence and Rigid Motion
LEssON 13: usiNg cONgruENcE critEria
3.
ExErcisEs
A
B
D
4.
C
ABCDE is a regular pentagon, with center point P.
B
C
A
P
D
E
5.
M is the midpoint of line segment AB.
C
A
Copyright © 2014 Pearson Education, Inc.
B
M
33
High School: Congruence and Rigid Motion
LEssON 13: usiNg cONgruENcE critEria
ExErcisEs
6. ∠ABC = 60°
A
60°
C
B
D
challenge Problem
7.
ACBDEFGH is a regular cube. The green triangle ABD is on the base of the cube, and the
orange triangle DBH goes through the interior of the cube. Use triangle congruence
criteria to show whether or not the two triangles are congruent.
H
G
E
F
D
A
Copyright © 2014 Pearson Education, Inc.
C
B
34
High School: Congruence and Rigid Motion
LEssON 14: sYMMEtriEs Of POLYgONs
ExErcisEs
ExErcisEs
For each figure, determine the number of lines of symmetry and the order of
rotational symmetry.
1.
2.
3.
Copyright © 2014 Pearson Education, Inc.
35
High School: Congruence and Rigid Motion
LEssON 14: sYMMEtriEs Of POLYgONs
4.
5.
6.
Copyright © 2014 Pearson Education, Inc.
36
ExErcisEs
High School: Congruence and Rigid Motion
LEssON 14: sYMMEtriEs Of POLYgONs
ExErcisEs
challenge Problem
7.
Copyright © 2014 Pearson Education, Inc.
Determine how many planes of symmetry and what order or rotational symmetry
this 3-D object has. It is a regular tetrahedron; a real life example is a 4-sided number
die. Think carefully about what rotational and reflective symmetry might mean in 3-D
space.
37
Math high School
congruence and
rigid Motion
anSWerS
exerciSeS
For exerciSeS
High School: Congruence and Rigid Motion
LEssON 2: rEfLEctiONs
ANSWERS
aNswErs
1.
The reflected images are shown:
2.
BOB BOX CEDE have a horizontal line of symmetry.
TOT MUM YAY have a vertical line of symmetry.
Copyright © 2014 Pearson Education, Inc.
40
High School: Congruence and Rigid Motion
LEssON 2: rEfLEctiONs
ANSWERS
3.
The resulting triangle will be Isosceles.
4.
B
y
5
B
4
3
2
1
–4
–3
–2
–1
1
2
3
4
5
x
–1
–2
–3
A
–4
–5
5.
The line of symmetry is y = x + 2.
y
9
8
7
6
5
4
3
2
1
–6 –5 –4 –3 –2 –1
–2
–3
–4
Copyright © 2014 Pearson Education, Inc.
1 2 3 4 5 6 7 x
41
High School: Congruence and Rigid Motion
LEssON 2: rEfLEctiONs
6.
ANSWERS
The original figure (yellow) is an obtuse triangle, the reflected image is in purple.
y
8
7
6
5
4
3
2
1
–1
–2
–3
–4
–5
–6
–7
1 2 3 4 5 6 7 8 9 x
challenge Problem
7.
The resulting graph, with all negative portions of the graph reflected up, will look like
this.
y
10
9
8
7
6
5
4
3
2
1
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1
–2
–3
–4
–5
–6
–7
–8
–9
–10
Copyright © 2014 Pearson Education, Inc.
1 2 3 4 5 6 7 8 9 10 x
42
High School: Congruence and Rigid Motion
LEssON 3: rOtatiONs
ANSWERS
aNswErs
1.
Here is the resulting figure:
y
6
5
4
3
2
1
1 2 3 4 5 6 7 x
–7 –6 –5 –4 –3 –2 –1
–2
–3
–4
–5
–6
2.
C Reflections
The reflections are across the line x = 2.
y
6
4
2
–6
–4
x=2
–2
2
–2
reflection
4
6
x
rotation
–4
–6
3.
You can see that the object passes through the points (0, 0), (5, 0), (5, 5), and (0, 5),
which if connected would create a perfect square inside the circle. Rotating the object
clockwise 90º would place point A at (5, 0). Reflecting it across the y-axis would place
A at (–5, 0). The distance between (–5, 0) and (5, 5) is:
( 5 − ( −5))2 + ( 5 − 0 )2
Copyright © 2014 Pearson Education, Inc.
= 125
43
High School: Congruence and Rigid Motion
LEssON 3: rOtatiONs
4.
ANSWERS
A
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
4
6
8
10 x
–2
–4
–6
–8
–10
5.
The following graph shows the original figure in green, and the rotated image figure in
orange.
y
6
4
2
–6
–4
–2
2
4
–2
–4
–6
Copyright © 2014 Pearson Education, Inc.
44
6
x
High School: Congruence and Rigid Motion
LEssON 3: rOtatiONs
ANSWERS
challenge Problem
6.
Answer will vary depending with the figure chosen. In this example, the figure ABCD is
rotated 90º around the point (3, –2) to get A'B'C'D. This figure is then reflected over the
x-axis to get A"B"C"D".
y
10
8
B
6
C"
A4
2
–12 –10 –8
–6
–4
–2
–2
C A'
D"
2
4
6
8
D
–8
–10
Copyright © 2014 Pearson Education, Inc.
45
10
A"
–4
–6
B"
C'
12 x
B'
High School: Congruence and Rigid Motion
LEssON 4: traNsLatiONs
ANSWERS
aNswErs
1.
D T(x, y) = (x + h, y + 3h)
2.
To shift the figure entirely into the fourth quadrant, you need to shift the figure so that
point A is below the x-axis, and point B is to the right of the y-axis. Any translation
T(x, y) = T(x + h, y + k) with x > 6 and y < –4 brings the entire figure into quadrant IV.
T(x, y) = (x + 7, y – 5) is one example.
3.
E'F'G' is the translated image, as shown in the following graph:
y
10
8
6
4
2
–10
–8
–6
–4
E'
–2
4 G' 6
2
–2
–4
10 x
8
E
F'
G
–6
F
–8
–10
4.
Copyright © 2014 Pearson Education, Inc.
The transformation is T(x, y) = (x + 5, y + 6). This can be determined by comparing one
point from the original and its image point, such as A(–6, 2) and A'(–6, 2). The other
points can be used to verify that the translation is correct.
46
High School: Congruence and Rigid Motion
LEssON 4: traNsLatiONs
5.
ANSWERS
Here is the original and translated figure. The translation is the red figure.
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
–2
–4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
47
4
6
8
10 x
High School: Congruence and Rigid Motion
LEssON 4: traNsLatiONs
ANSWERS
challenge Problem
6.
The figure I created ABCD has vertices at the following coordinates:
(–4, 3), (–3, 0), (4, 2), and (–1, 6).
After the translation T(x, y) = (x – 8, y + 4), the figure A'B'C'D' has the new coordinates
(–12, 7), (–11, 4), (–4, 6), (–9, 10). Both figures are shown on the following graph:
y
12
D'
10
8
A'
C'
B'
D
6
4
A
C
2
–14
Copyright © 2014 Pearson Education, Inc.
–12
–10
–8
–6
–4
B –2
48
2
–2
4
x
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
ANSWERS
aNswErs
1.
The results of the vertical stretch are shown. The new points are (–4, 8), (3, 6), (–2, –6),
and (3, –4). The grey figure is the resulting image.
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
2
4
6
10 x
8
–2
2
–44
–6
–8
–10
2.
y
10
8
6
4
2
–4
–2
2
4
6
8
–2
–4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
49
10
122
14
4
16
18
2200
22
24
x
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
ANSWERS
y
3.
10
8
6
4
2
–10
–8
–6
–4
–2
2
4
6
8
10 x
–2
–4
–6
–8
–10
4.
The shape I started with was a parallelogram. It has the coordinates (–2, 2), (3, 5), (–2,
–3), and (3, 0).
The yellow figure is an image after a translation (a rigid motion) of 6 units to the right.
The green figure is an image after a dilation (a non-rigid motion) centered at the
origin with a scale factor of 2.
Copyright © 2014 Pearson Education, Inc.
50
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
ANSWERS
y
10
8
6
4
2
–10
–8
–6
–44
–2
2
2
4
6
8
10 x
–2
2
–44
–6
–8
–10
You can make many observations about these two transformations. The translation
preserved all side lengths, while the dilation made them longer. The translation
preserved the shape’s area, while the dilation made it larger. The two sets of parallel
lines were preserved in
both cases.
5.
From the original figure, first I did the horizontal stretch, with a scale factor of 3. The
orange figure is the image after the stretch.
y
10
8
6
4
2
–10
–8
–6
6
–44
–2
–2
2
–2
2
–4
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
51
4
6
8
10 x
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
ANSWERS
Then I performed only the vertical translation, I used a shift of
5 units downward.
y
10
8
6
4
2
–10
–8
–6
–4
–2
2
2
4
6
8
10 x
–2
2
–44
–6
6
–8
8
–10
And then I made the image after both transformations, one after
the other.
y
10
8
6
4
2
–10
–8
–6
–4
–2
–2
2
4
6
8
10 x
–2
2
–44
–6
6
–8
8
–10
Here is a graph that includes all three images for comparison.
Copyright © 2014 Pearson Education, Inc.
52
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
ANSWERS
y
10
8
6
4
2
–10
–8
–6
6
–44
––2
2
2
4
6
8
10 x
–2
2
–44
–6
6
–8
8
–10
Again, we can make many observations about the different types of transformations. In
this case, we can also see that the set of parallel lines remained parallel even after the
horizontal stretch (although they both have new slopes, they both have the same new
slope) It is also interesting to note that it doesn’t matter which order you do the two
transformations in, you will end up with the same final image.
Copyright © 2014 Pearson Education, Inc.
53
High School: Congruence and Rigid Motion
LEssON 5: BEiNg PrEcisE
ANSWERS
challenge Problem
6.
The starting points are (–4, 1), (1, 1) (–1, –2,) and (–3, –1). It may be easier to just use
algebra to determine the new coordinates, and then plot the 4 new points.
T(x, y) = (3x + 2, 4y – 3)
So the new points will be (–10, 1), (5, 1,) (–1, –11) ,and (–7, –7).
The grey figure shows the transformed image, and the red figure shows the original:
y
12
10
8
6
4
2
–10
–8
8
–6
6
–44
–2
2
2
4
6
8
10 x
–2
2
–44
–6
6
–8
8
–10
10
The resulting motion is definitely non-rigid. It has changes the edge lengths and area
a great deal. This transformation is a mix of stretches and translations. It stretches
the horizontal coordinates by a factor of 3, and stretches the y-coordinates by a
factor of 4. It also shifts the (resulting) x-coordinates 2 to the right, and the resulting
y-coordinates 3 downward.
Copyright © 2014 Pearson Education, Inc.
54
High School: Congruence and Rigid Motion
LEssON 6: cOMPOsitE traNsfOrMatiONs
ANSWERS
aNswErs
1.
The orange figure shows the image after just the translation, then the purple figure
shows the image after the translation and rotation.
y
C'
10
D'
8
A'
B'
6
4
2
C
–10
–8
–6
–4
–2
2
–2
6
B
–6
–8
–10
55
8
D
A
–4
Copyright © 2014 Pearson Education, Inc.
4
10
12 x
High School: Congruence and Rigid Motion
LEssON 6: cOMPOsitE traNsfOrMatiONs
2.
ANSWERS
I used a translation and then a rotation to get the figure into
quadrant II. First, the translation moves the figure 5 units to the left (the red image)
and then the rotation rotates it –90º (clockwise) into quadrant II (the green image).
y
10
8
E"
6
F"4
2
G"
–10
–8
–6
–4
–2
–2
–4
F
2
4
6
8
10 x
G
–66
E
–8
–10
3.
There are many possible solutions. As long as the final image has the same coordinates
at A"B"C" in the original graph, the solution is valid.
I also used a translation, and then a rotation, but both different from the original
transformations. First, I translated the figure up 8 units. Then rotated the figure 180º
around the point (–3, 4), which is the center of the second image. This results in the
exact same A"B"C" image.
4.
There are many possible sequences of transformations, but the shortest sequence is
two steps. Here is one possible sequence of transformations.
First, rotate the figure 180º around the origin. R(0, 0), 180°.
Second, translate the figure 3 units up and 3 units right. T(x, y) = (x + 3, y + 3).
5.
One compound rotation of 170º will result in the same final image. To determine this,
simply add up all of the degree measures of each individual rotation.
6.
One compound translation up 3 units, and right 1 unit, will take DEF to D""E""F"". This
is determined by the net change done by all of the individual translations.
Copyright © 2014 Pearson Education, Inc.
56
High School: Congruence and Rigid Motion
LEssON 6: cOMPOsitE traNsfOrMatiONs
ANSWERS
challenge Problem
7.
Here is the final image after all three transformations:
y
10
8
6
G
H
I"'
H"'
4
2
G"'
J"'
–10
–8
–66
–4
4
I
–2
2
4
6
8
10 x
–2
J
–4
–6
–8
–10
Interestingly, the order matters quite a bit, here is an example where changing the
order of the transformations results in a totally different final image.
1. Rotate –90º around the origin to the red figure.
2. Reflect across the x-axis to the yellow figure.
3. Translate 7 units to the right to the purple figure.
y
10
8
6
G
H
4
2
–10
–8
–66
J
–4
4
I
–2
2
4
–2
–44
–66
–8
–10
Copyright © 2014 Pearson Education, Inc.
57
6
8
10
12 x
High School: Congruence and Rigid Motion
LEssON 11: dEfiNiNg cONgruENcE
ANSWERS
aNswErs
1.
The two triangles are indeed congruent.You can translate ABC up 2 units and right 3
units to map directly onto DEF.
2.
2. These two triangles are not congruent.You can show this by attempting to map ABC
onto DEF, but there is no way to do it with only rigid motions.You can get close by
reflecting ABC over the x-axis, but here is the resulting image, which shows that you
cannot map the triangles exactly onto each other.
y
6
B' E
D
–6
F
2
–4
–2
A
2
6 x
4
–2
C
–4
B
3.
4
–6
JKL and MNO are congruent. There are a number of different sequences of rigid
motions that prove this. One such method is to rotate 180º around the origin and then
translate 4 units down. The dotted triangle is JKL after the first transformation
(the rotation).
y
4
J
2
–8
–6
K
–4
–2
L
2
–2
4
6
8
N
–4
O
–6
M
–8
Copyright © 2014 Pearson Education, Inc.
58
x
High School: Congruence and Rigid Motion
LEssON 11: dEfiNiNg cONgruENcE
ANSWERS
4.
∆ABC and ∆DBC are indeed congruent.You can determine that each pair of sides and
each pair of angles is congruent, therefore the two triangles are congruent. AB = BD
and AC = DC. BC is shared by both triangles. Therefore, ∆ABC and ∆DBC are congruent.
5.
First determine the coordinates of each point. This will lead to finding the distances
between each vertex and angles between each side.
D = (–5, 2), E = (–3, 1), F = (–1, 5) and G = (5, –4), H = (3, –5), I = (1, –1)
You can determine the distance between each point to determine if the matching pairs
of sides are equal length.
DE ≅ GH = 2.24 units
EF ≅ HI = 4.47 units
DF ≅ GI = 5 units
You can also determine the angles by comparing slopes of the sides. For example, you
can show that both triangles are right triangles by comparing the slope of DE to EF
and the slope of GH to HI.
1
, and the slope of EF is 2. Since these are negative reciprocals of
2
each other, the angle they make must be 90°.
The slope of DE is −
1
Similarly, the slope of GH is , and the slope of HI is –2, these are also negative
2
reciprocals of each other, so the angle they make must be 90°.
All that is left is to show one more pair of angles (the third pair would solve itself once
we know the second pair) you can measure it using a protractor or an electronic
graphing tool.You will find that
∠FED ≅ ∠IGH = 63.43° and ∠DFE ≅ ∠GIH = 26.57°
6.
∆JKL is not congruent to ∆MNO. There are many ways to show this. As long as you
show one of the corresponding pairs of sides or angles does not match, this proves the
triangles are not congruent.
The length of JK is 4 units, the length of MN is 4.47 units, found by using the
Pythagorean theorem between points M and N. Since these matching sides do not have
the same length, the triangles are not congruent.
Copyright © 2014 Pearson Education, Inc.
59
High School: Congruence and Rigid Motion
LEssON 11: dEfiNiNg cONgruENcE
ANSWERS
challenge Problem
7.
Here is one pair of congruent triangles that meets the criteria. These were found by
graphing a right triangle in the third quadrant, and then rotating it 180º into the first
quadrant. It is certain that they are congruent, since the rigid motion of the rotation
links the two.
y
10
8
6
F
4
2
D
–10
–8
–6
B
–4
–2
A–2
E
2
4
–4
C
–6
–8
–10
Copyright © 2014 Pearson Education, Inc.
60
6
8
10 x
High School: Congruence and Rigid Motion
LEssON 12: triaNgLE cONgruENcE critEria
ANSWERS
aNswErs
1.
The triangles are congruent by SAS. BC and CD are given as both equal to 5 cm.
The 90° angles are congruent, and both triangles share the side CA, which must be
congruent to itself. This shows that ∆ABC and ∆ADC are congruent.
2.
The triangles are not congruent. Although they share sides that are all radii of the
circle, the given segment lengths of AB = 5 cm and DE = 8 cm make it impossible for
the triangles to be congruent.
3.
The triangles are congruent by ASA. The given 40° and 22° angles match on both
triangles, ∠ABC matches∠DCB and ∠ACB matches with ∠DBC. Also the two triangles
share the side BC, which must be congruent to itself. Therefore ∆ABC and ∆DCB are
congruent.
4.
The triangles are not necessarily congruent. Although the 120° angles match, and two
of the side lengths are given to match, the position of the matching pieces does not fit
any of the congruence criteria. This set up is an example of SSA, where the congruent
sides are adjacent, but the congruent angle is not in between them.You could
conceivably make different triangles using the given information.
5.
The triangles are congruent by SSS. All three sides are given as congruent lengths,
therefore the triangles must be congruent.
6.
Knowing AC ≅ DF shows congruence by SSS, and knowing ∠ABC = ∠DEF shows
congruence by SAS.
Copyright © 2014 Pearson Education, Inc.
61
High School: Congruence and Rigid Motion
LEssON 12: triaNgLE cONgruENcE critEria
ANSWERS
challenge Problem
7.
You could show the congruence using either method, the simpler one might be to
show that ABCD can be translated 5 units to the right to map directly onto EFGH. The
coordinates are as follows:
A = (2, 4), B = (4, 3), C = (4, 0), D = (2, –1)
By adding 4 to every x-coordinate, you get the exact coordinates of EFGH:
E = (7, 4), F = (9, 3), G = (9, 0), H = (7,–1)
You could also use the coordinate plane to help determine all of the side lengths and
angles in order to prove the congruence.
AD ≅ EH = 5 units
BC ≅ FG = 3 units
AB ≅ EF = 5 units
CD ≅ GH = 5 units
Then by comparing slopes, using circles or a graphing program, you can measure each
angle, since you know all of the coordinates.
∠DAB ≅ ∠HEF = 63.43°
∠CDA ≅ ∠GHE = 63.43°
∠ABC ≅ ∠EFG = 116.577 °
∠BCD ≅ ∠FGH = 116.57 °
Copyright © 2014 Pearson Education, Inc.
62
High School: Congruence and Rigid Motion
LEssON 13: usiNg cONgruENcE critEria
ANSWERS
aNswErs
1.
The two triangles are congruent, by ASA. AC is a transversal that cuts through
both pairs of parallel lines in the figure. This creates opposite interior angles at
∠CAB ≅ ∠ACD and at ∠DAC ≅ ∠BCA . Also the two triangles share the segment AC,
which is in between the two known angle congruence pairs. AC must be congruent to
itself, so the triangles are congruent.
2.
The triangles are congruent by SAS.You know all four segments from the center
to the circle are congruent, since they are all radii of the circle. This means that
CA ≅ CB ≅ CE ≅ CD .
Since BE is a diameter, you know it cuts a line exactly through the center of the circle.
So ∠BCE must be 180º, the given angles make supplementary angles with ∠BCA and
∠ECD, so by a transitive relationship ∠BCA ≅ ∠ECD .
CB ≅ CD, ∠BCA ≅ ∠ECD , and CA ≅ CE , therefore the triangles are congruent by SAS.
3.
The triangles are not congruent.You can get two congruent pieces, with the given
angles ∠DAC ≅ ∠CAB , and the shared side length AC. But this is not quite enough
information to conclude that the triangles are congruent.
4.
The triangles are congruent by SAS, or by SSS.You know each segment from the
center P to a vertex of the pentagon will be congruent, and you know each edge of
the pentagon is congruent as well. This gives you PC ≅ PB ≅ PD ≅ PE and also BC ≅ DE
which shows congruence by SSS.You can also use the interior angles of the pentagon
to show congruence. Since it is a regular pentagon, you know all of the interior angles
are congruent, so ∠CPB ≅ ∠DPE . This shows congruence using SAS.
5.
The triangles are congruent by SAS. Since M is the midpoint of AB, you know
AM ≅ BM . Since ∠BMC and ∠AMC are complementary and ∠BMC is given as a right
angle, ∠BMC ≅ ∠AMC. This is sufficient to say the triangles are congruent by SAS.
6.
The triangles are not congruent. They share the side length BC, and the ∠ABC is
given, but there is no other information to add to this. The 60° angle is not duplicated
anywhere since no lines are specified as parallel.
Copyright © 2014 Pearson Education, Inc.
63
High School: Congruence and Rigid Motion
LEssON 13: usiNg cONgruENcE critEria
ANSWERS
challenge Problem
7.
The triangles are not congruent. Say that the side length of the cube is 1. The base
∆ABD can then be drawn as:
D
45º
2
1
A
90º
45º
1
B
The orange triangle can be drawn in two-dimensional space. The base of triangle DBH
shares the longer side of triangle ABD. Segment DB is in both triangles, but they are
not corresponding sides. Using the Pythagorean theorem, you can fill in the hypotenuse
of the triangle DBH.
H
3
1
D
90º
2
B
The triangles are not congruent.
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High School: Congruence and Rigid Motion
LEssON 14: sYMMEtriEs Of POLYgONs
ANSWERS
aNswErs
1.
4 lines of symmetry
Order 4 rotational symmetry
2.
4 lines of symmetry
Order 4 rotational symmetry
3.
2 line of symmetry
Order 2 rotational symmetry
4.
1 line of symmetry
Order 1 rotational symmetry
5.
6 lines of symmetry
Order 6 rotational symmetry
6.
3 lines of symmetry
Order 3 rotational symmetry
challenge Problem
7.
One way to think about planes of symmetry is to imagine a 2-D mirror slicing through
a 3-D figure. If the reflection in the mirror completes the figure, then that is a plane
of symmetry. A regular tetrahedron has three planes of symmetry, each bisecting a
side and also going through the vertex opposite that side. Students should discuss and
debate their methods for analyzing the 3-D shape, here is one example analysis:
There are 7 axes of symmetry. 4 of them connect each vertex to the center of the
opposite face. The figure could rotate around each of these axes three times to map to
itself.
3 of them connect the midpoints of opposite edges. The figure is symmetric across the
planes that cut through the figure this way.
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