Chapter 7 Notes

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Chapter 7 Notes
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7.1 Rigid Motion in a Plane
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• Preimage, Image
• P P
• GP ? P notation in relation with functions.
• If distance is preserved, its an isometry.
• It also preserves angle measures, parallel lines,
  and distances. These are called rigid
  transformations.
• Example, shifting a desk preserves isometry. A
  projection onto a screen normally doesnt (it
  makes the lengths longer).

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Reflections
m
When a transformation occurs where a line acts
like a mirror, its a reflection.
P
P
Q
Q
RR
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Translations
When a transformation occurs where all the points
glide the same distance, it is called a
TRANSLATION.
P
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Rotations. A rotation is a transformation where
an image is rotated about a certain point.
P
O
P
Positive, counterclockwise Negative, clockwise
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Well describe the transformations
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Well describe the transformations and line up
some letters.
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You can show that something is an isometry on a
coordinate plane by using distance formula. Show
using distance formula which transformations are
isometric and which arent.
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7.2 Reflections
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Reflections
m
When a transformation occurs where a line acts
like a mirror, its a reflection.
P
P
Q
Q
RR
A reflection in line m maps every point P to
point P such that 1) If P is not on m, then m
is the perpendicular bisector of PP 2) If P is
on line m, then PP
Line of Reflection
Notation Rm P ? P
Name of line the transformation is reflecting
with.
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Write a transformation that describes the
reflection of points across the x-axis
Rx-axis(x,y) ? (x, -y)
Ry-axis(x,y) ? (-x, y)
Sometimes, they want you to reflect across other
lines, so you just need to count.
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Theorem 14-2 A reflection in a line is an
isometry. Therefore, it preserves distance,
angle measure, and areas of a polygon.
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Key to reflections is perpendicular bisectors.
You will need to construct in your homework, this
is how. Use construction to reflect PQ across
line m
Construct line perpendicular to m from point P
m
Use compass, intersection as center, swing
compass to other side. Make dot.
P
Q
Repeat, then connect dot.
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Sketch a reflection over the given line.
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Hit the black ball by hitting it off the bottom
wall.
Use reflection!
AIM HERE!
Reflection is an isometry, so angles will be
congruent by the corollary, so if you aim for the
imaginary ball that is reflected by the wall, the
angle will bounce it back towards the target.
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This concept also occurs in the shortest distance
concept
Where should the trashcan be placed so its the
shortest distance from the two homes.
Longer distance total!
HERE!
Shortest distance is normally a straight line, so
you want to mark where the shortest path would be
from the two different homes by using reflection.
Anywhere else will give you a longer path
(triangle inequality theorem).
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A figure in the plane has a line of symmetry if
the figure can be mapped onto itself by a
reflection in the line. We think of it as being
able to cut things in half.
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Sketch and draw all the lines of symmetry for
this shape
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7.3 Rotations
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Rotations. A rotation is a transformation where
an image is rotated about a certain point.
RO, 90P ? P
P
O
Amount of rotation.
Point of rotation
P
Fancy R, rotation
Positive, counterclockwise Negative, clockwise
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As you may know, a circle is 360 degrees, so if
an object is rotated 360, then it ends up in the
same spot.
RO, 360P ? P
P
O
P
Then P P
Likewise, adding or subtracting by multiples of
360 to the rotation leaves it at the same spot.
RO, 780P ? P
RO, -300P ? P
RO, 60P ? P
O
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A rotation about point O through xo is a
transformation such that 1) If a point P is
different from O, then OPOP and 2) If point P
is the same as point O, then P P
P
O
Thrm A rotation is an isometry.
P
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Find image given preimage and rotation, order
matters
RJ, 180ABJ ? RN, 180ABJ ? RN, 90IJN ? RN,
-90IJN ? RNFIMH ? RD, -90FND?
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Figure out the coordinate.
O Origin RO, 180P ? P
O Origin
RO, 90(2 , 0) ? ( ) RO, -90(0 , 3) ? (
) RO, 90(1 , -2) ? ( ) RO, -90(-2 , 3)
? ( ) RO, 90(x , y) ? ( ) RO, -90(x
, y) ? ( )
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Rotate point P 90 degrees clockwise.
RO,-90P ? P
P
O
P
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RmP?P RnP?P
P
The composite of the two reflections over
intersecting lines is similar to what other
transformation?
P
P
yo
O
n
m
Theorem A composite of reflections in two
intersecting lines is a rotation about the point
of intersection of the two lines. The measure of
the angle of rotation is twice the measure of the
angle from the first line of reflection to the
second.
Referencing the diagram above, how much does P
move by?
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Find angle of rotation that maps preimage to image
18o
70o
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7.4 Translations and Vectors
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Translations
When a transformation occurs where all the points
glide the same distance, it is called a
TRANSLATION.
Notation T P ? P
Theorem A translation is an isometry.
T for translation
Generally, you will see this in a coordinate
plane, and noted as such T (x,y) ? (x h, y
k) where h and k tell how much the figure
shifted.
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We will take a couple points and perform T(x,y)
? (x 2, y 3)
T(2,3) ? ( __ , __ ) T( , ) ? (5, -1)
T(-3,0) ? ( __ , __ ) T( , ) ? (0, 1)
T (a, b) ? ( __ , __ ) T( , ) ? (c, d)
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Vectors
Any quantity such as force, velocity, or
acceleration, that has both magnitude and
direction, is a vector.
Vector notation. ORDER MATTERS!
B
Initial Terminal
Component Form
A
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Write in component form
C
B
D
A
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Translations
You could also say points were translated by
vector
Translate the triangle using vector
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Write the vector AND coordinate notation that
describes the translation




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RmP?P RnP?P
The composite of the two reflections over
parallel lines is similar to what other
transformation?
P
P
P
m
n
Referencing the diagram above, how far apart are
P and P?
Theorem A composite of reflections in two
parallel lines is a translation. The translation
glides all points through twice the distance from
the first line of reflection to the second.
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M and N are perpendicular bisectors of the
preimage and the image. How far did the objects
translate ABC translated to ___________
m
n
------4.2 in --------
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7.5 Glide Reflections and Compositions
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A GLIDE REFLECTION occurs when you translate an
object, and then reflect it. Its a composition
(like combination) of transformations.
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We will take a couple points and perform T(x,y)
? (x 2, y 3) Ry-axisP ? P Then we will
write a mapping function G that combines those
two functions above.
T(2,3) ? ( __ , __ ) Ry-axis( __ , __ ) ? ( __
, __ )
T(-3,0) ? ( __ , __ ) Ry-axis( __ , __ ) ? ( __
, __ )
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Composites of mapping
Given transformations S and T, the two can be
combined to make a new transformation. This is
called the composite of S and T. You have
already seen an example of this in a GLIDE
REFLECTION.
Compost nation Your home for fertilizers.
Say T is translation two inches right
Happens Happens Second First Read S of T or
S after T
ORDER MATTERS!!!
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Composites of mapping
Given transformations S and T, the two can be
combined to make a new transformation. This is
called the composite of S and T. You have
already seen an example of this in a GLIDE
REFLECTION.
Say T is translation two inches right
Happens Happens Second First Read S of T or
S after T
ORDER MATTERS!!!
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Order matters in a composition of functions. The
composite of two isometries is an
isometry. There are times on a coordinate grid
where youll be asked to combine a composition
into one function, like you did for glide
reflections. There are also times when two
compositions may look like a type of one
transformation.
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Well do different combinations of
transformations and see what happens.
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Well do different combinations of
transformations and see what happens. Points and
Shapes. Also draw some transformations and
describe compositions.
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8.7 Dilations
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Dilations.
Center Scale factor
A dilation DO, k maps any point P to a point P,
determined as follows 1) If k gt 0, P lies on
OP and OP kOP 2) If klt0, P lies on the ray
opposite OP and OP KOP 3) The center is its
own image
P
O
k gt 1 is an EXPANSION, expands the picture k
lt 1 is a CONTRACTION, shrinks the picture
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A dilation DO, k maps any point P to a point P,
determined as follows 1) If k gt 0, P lies on
OP and OP kOP 2) If klt0, P lies on the ray
opposite OP and OP KOP 3) The center is its
own image
B
A
C
O
1) Draw a line through Center and vertex. 2)
Extend or shrink segment by scale factor.
(Technically by construction and common sense) 3)
Repeat, then connect.
k gt 1 is an EXPANSION, expands the picture k
lt 1 is a CONTRACTION, shrinks the picture
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A dilation DO, k maps any point P to a point P,
determined as follows 1) If k gt 0, P lies on
OP and OP kOP 2) If klt0, P lies on the ray
opposite OP and OP KOP 3) The center is its
own image
B
A
C
k gt 1 is an EXPANSION, expands the picture k
lt 1 is a CONTRACTION, shrinks the picture
O
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B
O
A
C
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When writing a scale factor of a dilation from O
of P to P, the scale factor is
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Identify scale factor, state if its a reduction
or enlargement (double checking), find unknown
variables.
O
6
2
B
30o
4
A
2x
C
Enlargement (this should match up with scale
factor)
B
4y
A
(2z)o
Find x, y, z
18
C
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Identify scale factor, state if its a reduction
or enlargement (double checking), find unknown
variables.
AA 2 AO 3
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A
B
2y
A
B
60o
(3z)o
x
O
12
C
D
C
D
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DO, 2
DO, 1/3
1) Dilate each point by scale factor and
label. 2) Connect
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• Find other sides, scale factor, given sides of
  one triangle, one side of another