What is similar about these objects? What do we need to pay attention to when objects are rotated?

What is similar about these objects?

What do we need to pay attention to when objects are rotated?

Course 2

8-10 Transformations

What am I learning today?Rotations

What will I do to show that I learned it?

Determine coordinates resulting from a rotation.

A full turn is a 360° rotation.

90°

180°

360°

How do you determine the How do you determine the angle of rotation?angle of rotation?

What are they rotating around?

270°

A quarter turn is a 90° rotation.

A half turn is a 180° rotation.

A three quarter turn is a 270° rotation.

Course 2

8-10 Rotations

QUESTIONWhat do I need to know to

complete a rotation?

To rotate:

Course 2

8-10 Rotations

- the direction – CW or CCW - the degrees – 90o, 180o, 270o

- the center or point of rotation – origin, vertex, or point inside the object

Course 2

8-10 Rotations

QUESTIONHow are objects rotated around the origin on a

coordinate plane?

Course 2

8-10 Rotations

To Rotate 180o around origin:1. Keep your x- and y-values the same..

2. Move to the opposite quadrant. I to III III to I II to IV IV to II.

3. Put the appropriate signs based on the quadrant.

Example:

Course 2

8-10 Rotations

Start: A (-4,3) in quadrant II

Rotate 180o clockwise

Finish: Quadrant IVx is positive andy is negative.

A’ (4,-3)

Course 2

8-10 Rotations

To Rotate 90o or 270o around origin:1. x- and y-value switch places. x becomes y and y becomes x..

2. Find the quadrant. Move one for 90o or three for 270o. Pay attention to the direction..

3. Put the appropriate signs based on the quadrant.

Example:

Course 2

8-10 Rotations

Start: A (-4,3) in quadrant II

Rotate 270o clockwise

Finish: Quadrant III x is negative and

y is negative.A’ (-3,-4)

Triangle ABC has vertices A(1, 0), B(3, 3), C(5, 0). Rotate ∆ABC 90° counterclockwise about the origin.

Rotations Around the Origin

Course 2

8-10 Rotations

x

y

A

B

C

3

–3

Graph the pre-image coordinates.

The coordinates of the image of triangle A’B’C’ are A’(0, 1), B’(-3,3), C (0.5).

Remember: A 90 degree rotation x and y change places, then pay attention to the

characteristics of the quadrants.

C’

B’

A’

Triangle ABC has vertices A(1, 0), B(3, 3), C(5, 0). Rotate ∆ABC 180° counterclockwise about the origin.

Rotations Around the Origin

Course 2

8-10 Rotations

x

y

A

B

C

3

–3

C’

B’

A’

Graph the pre-image coordinates.

The coordinates of the image of triangle ABC are A’(-1, 0), B’(-3,-3), C’(-5, 0).

Remember: A 180 degree rotation only changes the signs, so pay attention to the

characteristics of the quadrants.

Triangle ABC has vertices A(1, 0), B(3, 3), C(5, 0). Rotate ∆ABC 270° counterclockwise about the origin.

Rotations Around the Origin

Course 2

8-10 Rotations

x

y

A

B

C

3

–3

C’

B’

A’

Graph the pre-image coordinates.

The coordinates of the image of triangle A’B’C’ are A’(0,-1), B’(3,-3), C’(0,-5).

Remember: A 270 degree rotation x and y change places, then pay attention to the

characteristics of the quadrants.

Course 2

8-10 Rotations

QUESTIONHow are the coordinates

determined from a rotation around a vertex?

Course 2

8-10 Rotations

To Rotate around a vertex:1. Coordinates of the center of rotation

stay the same.2. Corresponding sides create an

angle equal to the degree of rotation

3. Each vertex in the shape must stay an equal distance from the center of rotation..

Triangle ABC has vertices A(-2,0), B(0, 3), C(0, –3). Rotate ∆ABC 90° clockwise about the vertex A.

Rotation around a vertex

Course 2

8-10 Rotations

x

y

B

C

3

-2

–3 B’

C’

A

The pre-image coordinates of triangle ABC are A(-2,0), B(0,3), C(0,-3).

The coordinates of the image of triangle ABC are A’(-2,0), B’(1,-2), C’(-5,-2).

The corresponding sides, AB and AB’ make a 90° angle.

Notice that vertex B is 2 units to the right and 3 units above vertex A, and vertex B’ is 3 units to the right and 2 units to the below vertex A.

Triangle ABC has vertices A(-2,0), B(0, 3), C(0, –3). Rotate ∆ABC 180° clockwise about the vertex A.

Rotation around a vertex

Course 2

8-10 Rotations

x

y

B

C

3

-2

–3B’

C’

A

The pre-image coordinates of triangle ABC are A(-2,0), B(0,3), C(0,-3).

The coordinates of the image of triangle ABC are A’(-2,0), B’(-4,-3), C’(-4,3).

The corresponding sides, AB and AB’ make a 180° angle.Notice that vertex B is 2 units to the right and 3 units above vertex A, and vertex B’ is 2 units to the left and 3 units below vertex A.

Where is the point of rotation?

A point OUTSIDE the shape A point INSIDE the Shape

How are these rotations similar?

How are these rotations different?

KK II MM

corresponding

rotation

Practice around OriginUsing these three points: P(6,3); C(-2,- 4); D(-1,0)

Rotate P 270o CCW

Rotate C 90o CW

Rotate D 180o CW

Rotate P 270o CW

Rotate C 180o CCW

Rotate D 90o CW

P’(3, -6)

C’(-4,2)

D’(1,0)

P’(-3,6)

C’(2,4)

D’(1,0)

Don’t forget to note:

What quadrant are you

starting in?

Practice around the Origin

P

R

Q

Rotate 90, 180, and 270 degrees counterclockwise

Practice around vertex C

A

CD

B

A

B

C

D A

BC

Rotate 90 degrees clockwise

Rotate 270 degrees counterclockwise

Rotate 180 degrees