Transcript Chapter 7 Notes

Chapter 7 Notes
7.1 – Rigid Motion in a Plane
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•
•
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Preimage, Image
P
P’
G:P  P’ notation in relation with functions.
If distance is preserved, it’s 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 doesn’t (it
makes the lengths longer).
Reflections
m
When a transformation
occurs where a line acts
like a mirror, it’s a
reflection.
P
Q
P`
R=R` Q`
Translations
When a transformation occurs where all the points ‘glide’ the
same distance, it is called a TRANSLATION.
P
(-5, 3)
(-3, 2)
(-4, 1)
Rotations.
A rotation is a transformation where an
image is rotated about a certain point.
P
O
P`
Positive, counterclockwise
Negative, clockwise
We’ll describe the transformations
We’ll describe the transformations and
line up some letters.
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 aren’t.
7.2 – Reflections
Reflections
When a transformation
occurs where a line acts
like a mirror, it’s a
reflection.
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`
m
P
P`
Q
R=R` Q`
Line of
Reflection
Notation
Rm: P  P`
2) If P is on line m, then
Name of line the transformation
P`=P
is reflecting with.
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)
(0, 5)
(-3,-1)
(1, -3)
Sometimes, they want you to reflect across other lines,
so you just need to count.
Theorem 14-2: A reflection in a line is an isometry.
Therefore, it preserves distance, angle measure, and
areas of a polygon.
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
Use compass,
intersection as center,
swing compass to other
side. Make dot.
Repeat, then connect
dot.
m
P
Q
Sketch a reflection over the given line.
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.
This concept also occurs in the shortest distance
concept
Where should the trashcan be placed so it’s
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).
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.
Sketch and draw all the lines of symmetry for this shape
7.3 – Rotations
Rotations.
A rotation is a transformation where an
image is rotated about a certain point.
RO, 90:P  P`
P
O
Amount of
rotation.
Point of
P`
rotation
Fancy R,
Positive, counterclockwise
rotation
Negative, clockwise
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, 360:P  P`
P`
Then P = P`
P
O
Likewise, adding or subtracting by multiples of 360
to the rotation leaves it at the same spot.
RO, 60:P  P` =RO, -300:P  P` =RO, 780:P  P`
O
A rotation about point O through xo is a
transformation such that:
1) If a point P is different from O, then
OP`=OP and mPOP ` x
2) If point P is the same as point O, then
P = P`
Thrm: A rotation is an
isometry.
P
O
xo
P`
Find image given preimage and rotation, order matters
RJ, 180:ABJ 
RN, 180:ABJ 
RN, 90:IJN 
RN, -90:IJN 
RNF:IMH 
RD, -90:FND
Figure out the
O = Origin
coordinate.
RO, 180:P  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)  (
)
Rotate point P 90
degrees clockwise.
RO,-90:P  P`
P
O
P`
Rm:PP`
Draw
Rn  Rm : P  P``
Rn:P`P``
P`
P
yo
P``
The composite of the two
reflections over intersecting
lines is similar to what other
transformation?
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?
Find angle of rotation that maps preimage to image
18o
70o
7.4 – Translations and Vectors
Translations
When a transformation occurs where all the points ‘glide’ the
same distance, it is called a TRANSLATION.
Notation
T: P  P`
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.
Theorem: A
translation is an
isometry.
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)
Vectors
Any quantity such as force, velocity, or acceleration,
that has both magnitude and direction, is a vector.
AB
Vector notation.
ORDER
MATTERS!
B
Initial
A
AB
Terminal
Component
Form
Write in component form
AB
CD
C
B
D
A
Translations
You could also say points
were translated by vector
Translate the triangle
using vector AB
AB  4,2
(-5, 3)
(-3, 2)
(-4, 1)
Write the vector AND coordinate notation that describes
the translation
‘
‘
‘
‘
Rm:PP`
P
m
Theorem
Draw
Rn  Rm : P  P``
Rn:P`P``
P``
P`
The composite of the two
reflections over parallel
lines is similar to what
other transformation?
n
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.
Referencing the diagram
above, how far apart are
P and P``?
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 --------
7.5 – Glide Reflections and
Compositions
A GLIDE REFLECTION occurs when you translate an object,
and then reflect it. It’s a composition (like combination) of
transformations.
We will take a couple points and perform:
T:(x,y)  (x + 2, y – 3)
Ry-axis:P  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:( __ , __ )  ( __ , __ )
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.
Compositionation:
n Notation
Compost
Your home for fertilizers.
S  T : P  P`` or S(T(P))  P`` or S  T : P  P``
Say T is translation two inches right
Happens
Second
Happens
First
Read “S of T” or “S
after T”
ORDER
MATTERS!!!
T  RO,-90 : P  P``
RO,-90  T : P  P``
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.
Compositio n Notation
S  T : P  P`` or S(T(P))  P`` or S  T : P  P``
Say T is translation two inches right
Happens
Second
Happens
First
Read “S of T” or “S
after T”
ORDER
MATTERS!!!
T  RO,-90 : P  P``
RO,-90  T : P  P``
Order matters in a composition of functions.
The composite of two isometries is an isometry.
There are times on a coordinate grid where you’ll 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.
We’ll do different combinations of transformations and see
what happens.
We’ll do different combinations of transformations and see
what happens. Points and Shapes. Also draw some
transformations and describe compositions.
8.7 – Dilations
DO ,k :
Dilations.
A dilation DO, k maps
any point P to a point
P`, determined as
follows:
1) If k > 0, P` lies on OP
and OP` = |k|OP
Center
Scale factor
P
O
2) If k<0, P` lies on the
ray opposite OP and
OP` = |K|OP
3) The center is its own
image
|k| > 1 is an EXPANSION,
expands the picture
|k| < 1 is a CONTRACTION,
shrinks the picture
DO,2 : ABC  A`B`C`
A dilation DO, k maps any
point P to a point P`,
determined as follows:
1) If k > 0, P` lies on OP and
OP` = |k|OP
B
2) If k<0, P` lies on the ray
opposite OP and OP` =
|K|OP
3) The center is its own
image
|k| > 1 is an EXPANSION,
expands the picture
A
O
C
1) Draw a line through Center and
vertex.
2) Extend or shrink segment by
|k| < 1 is a CONTRACTION, scale factor. (Technically by
construction and common sense)
shrinks the picture
3) Repeat, then connect.
D
A dilation DO, k maps any
point P to a point P`,
determined as follows:
1
O,
4
: ABC  A`B`C`
1) If k > 0, P` lies on OP and
OP` = |k|OP
B
2) If k<0, P` lies on the ray
opposite OP and OP` =
|K|OP
A
3) The center is its own
image
|k| > 1 is an EXPANSION,
expands the picture
|k| < 1 is a CONTRACTION,
shrinks the picture
C
O
D
1
O,
2
: ABC  A`B`C`
B
O
A
C
A dilationsometimesis isometricif | k | 1, but
it is ALWAYSmappinga similar figure. So a
dilationis called a SIMILARIT YMAPPING.
When writing a scale factor of a dilation from O of P to P’, the
scale factor is:
OP '
k
OP
Identify scale factor, state if it’s a reduction or enlargement
(double checking), find unknown variables.
O
2
30o
A
B
6
4
2x
C
B’
4y
A’
(2z)o
Enlargement (this
should match up with
scale factor)
Find x, y, z
18
C’
Set up your proportion like this
Image
k
Pre - Image
Identify scale factor, state if it’s a reduction or enlargement
(double checking), find unknown variables.
AA’ = 2
A’O = 3
20
2y
A
A’
12
x
B’
60o
(3z)o
O
D’
D
B
C’
C
DO, 2
DO, 1/3
1) Dilate each
point by scale
factor and
label.
2) Connect
• Find other sides, scale factor, given sides
of one triangle, one side of another