7.1 – Rigid Motion in a Plane

• 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

P

Q

P`

Q`

When a transformation occurs where a line acts like a mirror, it’s a reflection.

R=R`

m

Translations

When a transformation occurs where all the points ‘glide’ the same distance, it is called a TRANSLATION.

(-5, 3)

(-1, 5)

(-3, 2)

(1, 4)

(-4, 1)

(0, 3)P

P`

Rotations.

A rotation is a transformation where an image is rotated about a certain point.

Positive, counterclockwise

Negative, clockwise

OP

P`

We’ll describe the transformations

Rotation about a point (counterclockwise, positive)

Reflection

Translation

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

P

Q

P`

Q`

Line of Reflection

When a transformation occurs where a line acts like a mirror, it’s a reflection.

R=R`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 P`=P

Notation

Rm: P P`

m

Name of line the transformation is reflecting with.

R for reflection

Write a transformation that describes the reflection of points across the x-axis

Rx-axis:(x,y) (x, -y)

(1, -3)

(1, 3)

(-3,-1)

(-3, 1)

(0, 5)

(0, -5)

Ry-axis:(x,y) (-x, y)

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

mConstruct line perpendicular to m from point P

P

Q

Use compass, intersection as center, swing compass to other side. Make dot.

Repeat, then connect dot.

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.

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).

Longer distance total!

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`

Fancy R, rotation

Point of rotation

Amount of rotation.

Positive, counterclockwise

Negative, clockwise

OP

P`

RO, 360:P P`

As you may know, a circle is 360 degrees, so if an object is rotated 360, then it ends up in the same spot.

P

P`

Then P = P`

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

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

2) If point P is the same as point O, then P = P`

xPOPm `

OP

P`

xoThrm: A rotation is an isometry.

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

NIJ

EFL

FMN

BKN

DLE

BCD

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) ( )

Figure out the

coordinate.

O = Origin

RO, 180:P P`

Rotate point P 90 degrees clockwise.

RO,-90:P P`

O

P

P`

P

P`

P``

mn

Rm:PP` Rn:P`P``

``: PPRR

Draw

mn The composite of the two reflections over intersecting lines is similar to what other transformation?

A Rotation.

Referencing the diagram above, how much does P move by?

Twice the angle between m and n

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.

O

yo

signsWatch

2, yOmn RR R

Justify?

Find angle of rotation that maps preimage to image

18o

70o

36o

140o

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.

ABVector notation. ORDER MATTERS!

Initial Terminal

A

B

AB Component Form

3,7

Write in component form

A

B

AB 5,1

CD 3,4

D

C

Translations

Translate the triangle using vector

(-5, 3)

(-1, 5)

(-3, 2)

(1, 4)

(-4, 1)

(0, 3)

You could also say points were translated by vector

AB 2,4 AB

Write the vector AND coordinate notation that describes the translation

‘

‘ ‘

‘

Switch it now

<5,4>

(x + 5, y + 4)

<-5,-4>

(x – 5, y – 4)

P P` P``

m n

Rm:PP` Rn:P`P``

``: PPRR

Draw

mn The composite of the two reflections over parallel lines is similar to what other transformation?

A Translation.

Referencing the diagram above, how far apart are P and P``?

Twice the distance between m and n

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.

m n------4.2 in --------

M and N are perpendicular bisectors of the preimage and the image.

How far did the objects translate

ABC translated to ___________

8.4 inA”B”C”

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

P``P:TSor P``S(T(P))or P``P:TS

Notationn Compositio

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.

Happens Happens Second First

Read “S of T” or “S after T”

O

ORDER MATTERS!!!

P``P:

P``P:T

O,-90

O,-90

T

R

RSay T is translation two inches right

P

P``

Composites of mapping

P``P:TSor P``S(T(P))or P``P:TS

Notationn Compositio

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.

Happens Happens Second First

Read “S of T” or “S after T”

OORDER MATTERS!!!

P``P:

P``P:T

O,-90

O,-90

T

R

RSay T is translation two inches right

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.

Find image given preimage, order matters

8.7 – Dilations

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

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

P

O

:,kOD

Center Scale factor

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

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

A

O

```:2, CBAABCDO

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.

B

C

A`

B`

C`OBSERVE THE NEXT TWO SLIDES.

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

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

A

O

```:4

1,

CBAABCDO

B

CA`B`

C`

```:2

1,

CBAABCDO

AO

B

C

MAPPING. SIMILARITY a called isdilation

a So figure.similar a mapping ALWAYS isit

but 1,|k| if isometric is sometimesdilation A

When writing a scale factor of a dilation from O of P to P’, the scale factor is:

OP

OPk

'

Identify scale factor, state if it’s a reduction or enlargement (double checking), find unknown variables.

2 6

A

B

C

A’

B’

C’

32

6k

Enlargement (this should match up with scale factor)

2x

18

Find x, y, z

4y

4

(2z)o

30o

x = 3

y = 3

z = 15k

Image-Pre

Image

thislike proportionyour upSet

O

Identify scale factor, state if it’s a reduction or enlargement (double checking), find unknown variables.

A B

C

A’ B’

C’

x12

2y

20

(3z)o

DD’

O

AA’ = 2

A’O = 3

60o

DO, 1/3DO, 2

1) Dilate each point by scale factor and label.

2) Connect