Transcript Slide 1

Isometries Rotations Translations Compositions
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Isometries-100
What is the definition of an isometry?
Give three examples of isometries.
answer
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Isometries-100
Answer
question
Isometry: a transformation that perseveres
length, angle measure, parallel lines, etc.
ex. Reflections
Rotations
Translations
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Isometries-200
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Which of the following is not a rotation of
?
a)
b)
c)
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c)
Isometries-200
Answer
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Isometries-300
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True or false?
1) Transformations that are not isometries
are called rigid transformations.
2) Flips, turns and slides are nicknames for
reflections, rotations and translations
3) Isometries preserve angle measures and
parallel lines
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1. False
2. True
3. True
Isometries-300
Answer
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Isometries-400
Find the value of each variable if the given
transformation is an isometry
50°
4d
7
50°
2c+3
a°
b°
12
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a=90°
b=40°
180-90-50
90-50
40
Isometries-400
Answer
c=2
2c+3=7
2c=4
c=2
d=3
4d=12
d=3
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Isometries-500
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Is the given transformation an isometry?
ABC
XYZ
A=(-4,2) X=(2,2)
B=(-1,4) Y=(4,-1)
C=(-1,1) Z=(1,-1)
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Isometries-500
Answer
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Yes
Use the distance formula to compare the side lengths
AB=√(-4+1)²+(2-4)² BC=√(-1+1)²+(4-1)² AC=√(-4+1)²+(2-1)²
=√(-3)²+(-2)²
=√(0)²+(3)²
=√(-3)²+(1)²
=√9+4
=√9
=√9+1
=√13
=3
=√10
XY=√(2-4)²+(2+1)² YZ=√(4-1)²+(-1+1)² XZ=√(2-1)²+(2+1)²
=√(-2)²+(3) ²
=√(3)²+(0)²
=√(1)²+(3)²
=√4+9
=√9
=√1+9
=√13
=3
=√10
AB=XY
BC=YZ
AC=XZ
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Rotations-100
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Does this figure
have rotational
symmetry? If so,
describe the
rotation that maps
the figure onto
itself.
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Rotations-100
Answer
question
Yes, the star does
have rotational
symmetry. To map
the figure onto
itself, you could
rotate the object
72° or 144°.
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Rotations-200
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A=(2,-3) Al=(-3,-2)
If A was rotated clockwise around the
origin, what was the angle of rotation?
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Rotations-200
Answer
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90°
In a 90° clockwise rotation, (x,y) (y,-x)
If you use that information, you can
substitute in (2,-3) to get (-2,-3), which are
the coordinates of the given pre-image
and image
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Rotations-300
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m
K
.
A
.A
.A
l
138°
ll
What is the
measure of the
angle of
rotation?
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Rotations-300
Answer
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84 °
When you reflect a figure over line k then over line m, the
angle of rotation is 2x (x=the measure of the acute angle
formed by k and m)
So, x=180-138
x=42
2(42)=84°
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Rotations-400
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Rotate (7,-2) 90°clockwise around the
origin. Name the point of the image. Do
the same for 180° and 270° clockwise.
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Rotations-400
Answer
90°=(-2, -7) because (x,y) (y,-x)
180°=(-7,2) because (x,y) (-x,-y)
270°=(2,7) because (x,y) (-y,x)
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Rotations-500
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5c
4b
Find the
values of
all the
variables
10
65°
a°
8
5
2d+2
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a=130°
a=2(65)
a=130
b=2
4b=8
b=2
Rotations-500
Answer
c=1
5c=5
c=1
d=4
2d+2=10
2d=8
d=4
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Translations-100
Reflect AB, A=(3,-3) B= (2,-4), over y=1.
What are the coordinates of Al and Bl
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Translations-100
Answer
Al=(3,5) Bl=(2,-6)
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Translation-200
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Find the other endpoint using the following
vectors.
1.(-4,0) vector <2,-3>
2. (5, -2) vector <5,1>
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Translation-200
Answer
1.(-2,-3)
(-4+2,0-3)
(-2,-3)
2. (10,-1)
(5+5,-2+1)
(10,-1)
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Translation-300
Use the following coordinate
notation to find the other
endpoint.
(x, y) (x+2, y-3)
1.(1,4)
2. (-3, -1)
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Translation-300
Answer
1. (3,1)
(1+2,4-3)
(3,1)
2. (-1,-4)
(-3+2,-1-3)
(-1,-4)
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Translation-400
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A translation of AB is described by vector
PQ<2,-5>. Find the value of each
variable.
A(w-5,-3)Al(10,x-1)
B(z,3y+1)Bl(5,5)
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Translation-400
Answer
w=10
w-5+2=10
w-3=10
w=13
x=-7
-3-5=x-1
-8=x-1
-7=x
y=3
3y+1-5=5
3y-4=5
3y=9
y=3
z=3
z+2=5
z=3
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Translation-500
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Write the equation for the line of reflection
A=(2,3) B=(6,-1)
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Translation-500
Answer
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y= x-3
Explanation: (2,3) (6,-1)
Slope= (3+1)=-1
midpoint=(6+2 3-1)= (4,1)
(2-6)
2 , 2
Perp. Line slope=1
y=1x+b
1=1(4)+b
1=4+b
-3=b
y=x-3
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Compositions-100
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What is a composition? What is a glide
reflection?
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Compositions-100
Answer
question
A composition is when 2 or more transformations
are combined to form a single transformation
A glide reflection is a transformation in which every
point P is mapped onto P by the following 2 steps
-a translation that maps P onto P
-a reflection in line k such that the line of
translation is parallel to reflection line k
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l
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Compositions-200
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When you switch the order of transformations,
does it affect the final image? In what cases?
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Compostitions-200
Answer
question
In a composition, it does affect the final
image, but it does not in a glide reflection.
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Compostitions-300
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Rotate A(3,2) 90° about the origin and reflect
over the x-axis.
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Al (2,-3)
All(2,3)
Compositions-300
Answer
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Compositions-400
Sketch the image of AB, A(4,2) B(7,0),
after a composition using the given
transformations (in the given order)
Translation:
(x,y) (x-4,y+2)
Rotation:
270° clockwise about the origin
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Compositions-400
Answer
Translation:
A(0,4) B(3,2)
Rotation:
A(-4,0) B(-2,3)
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Compositions-500
Sketch the image of A (-5,2) after translating
it using vector <3,-4> and reflecting over
x=4
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Compositions-500
Answer
After translation: Al(-2,-2)
After reflection: All(10,-2)
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