Transformations Day 5 - Composite Transformations

and ConjecturesSeptember 29 2014

Agenda• Today, I will learn how to map equivalent transformations. I will

learn how to describe transformations in writing.

• Bellringer: There is an answer key for the last class’s worksheet on your desk. Swap papers and grade it using a colored pencil or pen with your table mates. (10 minutes)

• Composite Transformations Continued: Taking notes on Ms. Wittenberg’s Lecture (20 minutes)

• We do: Answering questions on the smart board (20 minutes)

• You do: Answering questions with your table mates (30 minutes)

• Exit Slip (10 minutes)

Intro• Let’s focus on a problem we looked at on our

worksheet:

• The vertices of triangle ABC are A(-4,4), B(-1,2) and C(-4, 1). Find the vertices of triangle ABC after this composite transformation (x,y) —> (x+2, y-2) then (x-1, y+4).

• What is one translation that could have given the same result as the composite transformation above?

Equivalent Transformations

• Let’s focus on a problem we looked at on our worksheet:

• The vertices of triangle ABC are A(-5,4), B(-1,2) and C(-5, 1). Find the vertices of triangle ABC after this composite transformation (x,y) —> (x+2, y-2) then (x-1, y+4).

• Let’s focus on a problem we looked at on our worksheet:

• The vertices of triangle ABC are A(-4,4), B(-1,2) and C(-4, 1). Find the vertices of triangle ABC after this composite transformation (x,y) —> (x+2, y-2) then (x-1, y+4).

Equivalent Transformations

• Let’s focus on a problem we looked at on our worksheet:

• The vertices of triangle ABC are A(-4,4), B(-1,2) and C(-4, 1). Find the vertices of triangle ABC after this composite transformation (x,y) —> (x+2, y-2) then (x-1, y+4).

Equivalent Transformations

• Let’s focus on a problem we looked at on our worksheet:

• The vertices of triangle ABC are A(-4,4), B(-1,2) and C(-4, 1). Find the vertices of triangle ABC after this composite transformation (x,y) —> (x+2, y-2) then (x-1, y+4).

• Is there a single transformation that we could have done instead of doing both?

• This is called an EQUIVALENT transformation

Equivalent Transformations

• Properties of Translation

• The lines connecting the pre-image to the image vertices of are all _________ and _________

• The pre-image is congruent to the image

A

B

C

Reflection Properties

A’

B’

C’

A

B

C

Reflection Properties lines connecting pre-image to image vertices are parallel

line of reflection is perpendicular bisector of lines connecting pre-image vertices to image vertices

A’

B’

C’

A

B

C

Reflection Properties

A

B

C

A

B

C

• Properties: • The lines connecting the pre-

image vertices to the image vertices are ____________ to the line of reflection.

• The line of reflection ____________ each of the purple lines (lines connecting pre-image to image points).

• The pre-image triangle is congruent to the image triangle.

• Each side of the pre-image triangle is a congruent segment to the corresponding side of the image triangle.

• Each of the purple lines is ___________ to each of the other purple lines

Properties of Reflections

Rotation Properties

• Image and Pre-Image are the same distance from the center of rotation

• Angles between Pre-Image and Image points through origin are all congruent

• Legs of each angle are congruent to each other

Rotation Properties

• You can see that the segments connecting the pre-image and image points are different lengths and intersecting.

Instead of me boring you with a whole lecture today. . .

• You’re going to do a tracing paper activity with your group!

• Groups 1-3 will complete the tracing paper activity on page 384 of Discovering Geometry

• Groups 4-6 will complete the tracing paper activity on page 385 of Discovering Geometry

• You have 3 minutes to complete the activity, then 7 minutes to discuss the reflection questions with your group.

• In 10 minutes we will come back together and teach each other what we’ve learned!

We Do 1Name the single translation that can replace the composition of these three translation rules (x+2, y+3), then (x-5, y+7), then (x+13, y-0)

We Do 2Name the single rotation that can replace the composition of these three rotations about the origin: 45°, then 50°, then 85°

We Do 3Lines m and n are parallel and 10 cm apart.

a. Point A is 6cm from line m and 16 cm from line n. Point A is reflected across line m, and then its image, A’, is reflected across line n to create a second image, point A’’. How far is point A from point A’’. b. What if A is reflected across n, and then its image is reflected across m? Find the new image and distance from A.