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Lesson 7.1 Rigid Motion in a Plane. Today, we will learn to… > identify the 3 basic transformations > use transformations in real-life situations. Transformations. The original figure is called the ____________ and the new figure is called the ____________. preimage. image. - PowerPoint PPT Presentation
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Lesson 7.1
Rigid Motion in a Plane
Today, we will learn to…> identify the 3 basic transformations> use transformations in real-life
situations
Transformations
The original figure is called the ____________ and the new figure is called
the ____________.
preimage
image
Transformations
Preimage: A , B , C , D
Image: A’ , B’ , C’ , D’
Rotation
Translation
Isometriespreserve length, angle measures,
parallel lines, & distances between points
Theorems 7.1, 7.2, & 7.4
Reflections, translations, and rotations are
isometries.
1. Name and describe the transformation.reflection over
ABC
the y-axis
A’B’C’
2. Name the coordinates of the vertices of the preimage and image.
(-4,0)
(-4,4)
(4,0)
(4,4)
(0,4)
3. Name and describe the transformation.reflection over
ABCD
x = -1
HGFE
4. Is the transformation an isometry? Explain.
NO
YES
YES
NO
5. The mapping is a reflection. Which side should have a length of 7?
Explain.WX = 7
6. Name the transformation. Find x and y.
Reflection
x = y = 40 4
Reflection
7. Name the transformation. Find x and y.
x = y = 12 4
8. Name the transformation. Find a, b, c, and d.
a =b =c =d =
7353158
reflection
9. Name the transformation. Find p, q, and r.
p =q =r =
193
7.5
rotation
10. Name the transformation and complete this statementGHI ____LKP
reflection
11. Name the transformation that maps the unshaded turtle onto the shaded turtle
reflection
translation
rotation
Lesson 7.2Reflections
Today, we will learn to…> identify and use reflections > identify relationships between
reflections and line symmetry
Reflection
2 images required
1. Is this a reflection?What is the line of reflection?
YES x = -2
2. Is this a reflection?
NO
3. Is this a reflection?What is the line of reflection?
YES
y = 1
4. Is this a reflection?What is the line of reflection?
YES
y = x
5. Is this a reflection?What is the line of reflection?
YES
y = - x
When can I use this in “Real Life?”
Finding a minimum distance
Telephone Cable - Pole PlacementTV cable (Converter Placement)
Walking Distances
Helps you work smarter not harder
Finding a minimum distance
6. A new telephone pole needs to be placed near the road at point C so that the length of telephone cable (AC + CB) is a minimum distance. Two houses are at positions A and B. Where should you locate the telephone pole?
A B
C
A’
Finding a minimum distance
1) reflect A2) connect A’ and B3) mark C
Line of Symmetry
1 image reflects onto itself
7. How many lines of symmetry does the figure have?
1 23
4
5
678
8. How many lines of symmetry does the figure have?
2
m A = can be used to calculate the angle between the mirrors
in a kaleidoscope
n = the number of lines of symmetry
180˚ n
1 23
4
5
6
78
180˚
8= 22.5˚
http://kaleidoscopeheaven.org
180˚
9= 20˚
10. Find the angle needed for the mirrors in this kaleidoscope.
180˚
4= 45˚
Project?1) Identify a reflection in a flag
2) Identify a line of symmetry
Reflection
Line of Symmetry
Reflection
Line of Symmetry
Section 7.2 Practice Sheet !!!
Lesson 7.3Rotations
students need tracing paper
Today, we will learn to…> identify and use rotations
Rotation
Angle of Rotation?
Center of Rotation?
Direction of Rotation?
Clockwise rotation
of 60°
Center of Rotation?
Angle of Rotation?
60˚60˚
Counter-Clockwise rotation
of 40°40°
40°
Theorem 7.3A reflection followed by a
reflection is a rotation.
If x˚ is the angle formed by the lines of reflection, then the angle of rotation is 2x°.
A
B’
A’
A’’
B’’
B 2x˚x˚
1. What is the degree of the rotation?
140˚
70˚
AA’
A’’
2. What is the degree of the rotation?
110˚
A A’
A’’
?125˚55˚
3. Use tracing paper to rotate ABCD 90º counterclockwise about the origin.
B (4, 1)Figure ABCD
Figure A'B'C'D'
A (2, –2)
A’ (2, 2) B ‘ (–1, 4)
C (5, 1)
C ‘ (–1,5)
D (5, –1)
D ‘(1, 5)
Rotational Symmetry
A figure has rotational symmetry if it can be mapped
onto itself by a rotation of 180˚ or less.
I had another dream….
6. Describe the rotations that map the figure onto itself.
= 45˚360˚
81
2
3
45
6
7
8
___ rotational symmetry 45˚
Describe the rotations that map the figure onto itself.
360 2
= 180˚ 1
2 ____ rotational symmetry 180˚
Describe the rotations that map the figure onto itself.
___ rotational symmetryno
Describe the rotational symmetry.
12
3
4
5
6
3606 = 60˚
60˚ rotational symmetry
Which segment represents a 90˚clockwise rotation of AB about P?
CD
Which segment represents a 90˚counterclockwise rotation of HI about Q?
LF
Project?1) Identify a rotation in a flag
2) Identify rotational symmetry in a flag
Rotation
60° Rotational symmetry
A
B
C
D
E
J
P KM
H F
G
L
Section 7.3 Practice!!!
Lesson 7.4Translations and
Vectors
Today, we will learn to…> identify and use translations
Translation
One reflection after another in two parallel lines creates a translation.
m n
THEOREM 7.5
PP '' is parallel to QQ''
k mQ
P
Q '
P '
Q ''
P ''
PP '' is perpendicular to k and m.______________
_______
k mQ
P
Q '
P '
Q ''
P ''
2d
d
The distance between P and P” is 2d, if d is the distance between
the parallel lines.
Name two segments parallel to YY”
XX”
ZZ”
Find YY”
6 cm
12 cm
XX”=
ZZ”=
12 cm
12 cm
A translation maps XYZ onto which triangle?
X”Y”Z”
Name two lines to XX”
line k
line m
A translation can be described by coordinate notation.
(x, y) (x + a, y + b) describesmovement
left or right
describesmovementup or down
(x, y) (x + 12, y - 20) means to translate the figure…
right 12 spaces & down 20 spaces
– + + –
1. (x, y) (x + 1, y – 9)
Use words to describe the translation.
2. (x, y) (x – 2, y + 7)
right 1 space , down 9 spaces
left 2 spaces, up 7 spaces
(x, y) (x + 5, y – 3)
3. left 5, down 10
Write the coordinate notation described.
4. up 6
(x – 5, y – 10)
(x, y + 6)(x , y)
(x , y)
5. Describe the translation with coordinate notation.
-2
+3
+3
-2
(x,y) (x – 2, y + 3)
6. Describe the translation with coordinate notation.
-7-2-2 -7
(x,y) (x – 7, y – 2)
-2 -7-2 -7
7. A triangle has vertices (-4,3);(0, 4); and (3, 2). Find the
coordinates of its image after the translation (x, y) (x + 4, y – 5)
(-4, 3) (-4 + 4, 3 – 5)
(3, 2) (3 + 4, 2 – 5)
(0, 4) (0 + 4, 4 – 5)
(7, -3)
(4, -1)
(0,-2)
Graphically, it would be…
(x, y) (x + 4, y – 5)
(-4, 3) (0, -2) (0, 4) (4, -1)
(3, 2) (7, -3)
preimage image (x, y) (x + 6, y – 2)
8. Find the image of (-4, 5)
9. Find the preimage of (9, 5)
(2, 3)
(3, 7)
(-4, 5) (-4 + 6, 5 – 2) ( __, __ )
( _ , _ ) ( x + 6, y – 2) ( 9, 5 ) x + 6 = 9 y – 2 = 5
A vector is a quantity that has both direction
and magnitude (size).
A vector can be used to describe a translation.
3 units up
5 units right
initial point
terminal pointBA
5
4
2 B
A
5 3,
The vector component form combines the horizontal and
vertical components.
5 3,
(x, y) (x + 5, y + 3)
Write this in coordinate notation form
4
2
-2
-4
-5 5
D'
C'B'
A'
D
B
A
C
10. What is the component form of the vector used for this translation?
4 2,
11. Name the vector and write its component form.
XY
X
Y5, 3
Write this in coordinate form.
(x,y) (x + 5, y – 3)
12) Describe the translation which maps ABC onto A’B’C’ by writing the translation in coordinate form and in vector component form.
A(3,6); B(1,0); C(4,8); A’(1,2); B’(-1,-4); C’(2,4)
(x, y) (x – 2, y – 4) – 2, – 4
Project?1) Identify a translation in a flag
Translation
Project?Two Objects
RequiredOne Object
OnlyReflection Line of Symmetry
RotationRotational Symmetry
Translation
Lesson 7.5Glide Reflections and
Compositions
students need worksheets and tracing paper
glide reflection
Example #1 Example #2
To be a “glide” reflection, the translation must be parallel
to the line of reflection.
NOTa glide reflection
NOTa glide reflection
These are just examples of a translation followed by a reflection.
Two or more transformations are combined to create a
composition.
A
A (2, 4) A’ ( , ) A’’ ( , )
1. translation: (x,y) (x, y+2) reflection: in the y-axis
2 6 -2 6
A’A”
2. reflection: in y = x translation: (x,y) (x+2, y-3)
A
A (-3, -2) A’ ( , ) A’’ ( , )
-2 -3 0 -6
A’A”
A
A”
B
A’
B’
B’’
A’’ (-1,- 4) and B’’ ( 2,- 1)
3. translation: (x,y) (x-3, y) reflection: in the x-axis
A (2, 4)
and B (5, 1)
4. translation: (x,y) (x, y+2) reflection: in y = -x
A (0, 4)
and
B (3, 2).
A’’ (-6, 0) and B’’ ( -4,-3)
BA B’A’
B”
A”
5. Describe the composition.
Reflection: in x-axisTranslation:(x,y) (x + 6,y + 2)
6. Describe the composition.
Reflection: in y = ½ Rotation: 90˚ clockwise about (1,-3)
PracticePractice Practice
How do we get better?