Translations & Vectors, Glide Reflections & Compositions

Objectives: Identify and use translations Identify glide reflections in a plane. Represent transformations as compositions of simpler transformations.

Translations

If I move a figure over without changing the size, angles, or orientation, that is a translation.

Translations

A translation is a transformation that maps every two points P and Q in the plane to points P’ and Q’ so that: PP’ = QQ’ PP’ || QQ’, or PP’ and QQ’ are collinear

PP’

QQ’

Translation Theorem

A translation is an isometry.

Another Theorem

Look at the picture at the bottom of p. 4212 reflections = a translation If lines k and m are parallel, then a reflection in

line k followed by a reflection in m is a translation.

If P” is the image of P, then the following is true:Line PP” is perpendicular to k and m.PP” = 2d where d is the distance between k &

m.

Using this theorem

In your book, look at the picture on the top of p. 422.

What segments are congruent?Does AC = BD?How long is GG”

Using this theorem

Look at the picture in the middle of p. 422.What is the horizontal shift?What is the vertical shift?Translations can be described as (x, y) --> (x+a, y+b)Where each point shifts a units

horizontally and b units vertically

Using this theorem

Look at the picture at the bottom of p. 422.What is the horizontal shift? -3What is the vertical shift? 4

Translations Using Vectors

A vector is a quantity that has both direction and magnitude (size) and is represented by an arrow drawn between 2 points.

3

5

Translations Using Vectors

The initial point of this vector is P and the terminal point is Q.

The component form of a vector combines the vertical and horizontal components. The component form of PQ is <5, 3>

3

5

P

Q

Identifying Vector Components

On p. 423, in the middle, what is the name of the vector in a.?

JKWhat is the component form?<3,4>On p. 423, in the middle, what is the name of the vector

in b.? What is the component form?MN <0,4>On p. 423, in the middle, what is the name of the vector

in c.? What is the component form?TS <3,-3>

Look at p. 423 Example 4

Translate ∆ABC using a vector of <4,2>The green arrows show each point of the

triangle going over 4 and up 2.The ends of these arrows create the new

triangle

Look at p. 424 Example 5

What is the component form of the vector that can be used to describe the translation?

Classwork & Homework

Classwork Page 425 16-34 evens

Homework 7.4 Practice B and C When you have finished the classwork you

can show me and I will give you the homework worksheets to work on

Glide Reflections & Compositions

A translation, or glide, and a reflection can be performed one after the other to produce a transformation known as a glide reflection.

For example, look at the bottom of p. 430.The blue to the red is (x,y) --> (x+10, y)The red to the green is a reflection about

the x axis.

Using Compositions

When 2 or more transformations are combined to produce a single transformation, the result is called a composition.

Composition Theorem

The composition of 2 or more isometries is an isometry.

For example:

If point P is at (2, -2) and point Q is at (3, -4)Sketch point PQRotate PQ 90˚ counterclockwise about the

origin.Reflect PQ in the y axis.Look at p. 431 to see this

If you switch the order . . .

And do the reflection first and the rotation second, how does it change the result?

See the picture at the bottom of p. 431.

Look at p. 432 at the top

What composition do you see?1. Reflection in the line x = 22. Rotation 90˚ clockwise about the point

2,0

Pentominoes

Look at the pentonimoes at the bottom of p. 432

Notice how we can use rotations, reflections and translations to move 2 of the pieces on the sides to fill the gaps

Do p. 435 #38

Classwork and Homework:

Classwork: 7.5 Practice AHomework: 7.5 Practice B & CWhen you finish Practice A show me and I

will give you B and C