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13.6 and 13.7. Rotations, Symmetry and Dilations. Goal Statement. w ill rotate figures and identify rotational symmetry w ill dilate figures in a coordinate plane. Can you identify the following terms?. r otation: center of rotation: angle of rotation:.

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13 6 and 13 7

13.6 and 13.7

Rotations, Symmetry and Dilations

goal statement
Goal Statement
  • will rotate figures and identify rotational symmetry
  • will dilate figures in a coordinate plane
can you identify the following terms
Can you identify the following terms?
  • rotation:
  • center of rotation:
  • angle of rotation:
can you identify the following terms1
Can you identify the following terms?
  • rotation: a transformation where a figure is turned about a fixed point
  • center of rotation:
  • angle of rotation:
can you identify the following terms2
Can you identify the following terms?
  • rotation: a transformation where a figure is turned about a fixed point
  • center of rotation: the fixed point (ex. 0,0)
  • angle of rotation:
can you identify the following terms3
Can you identify the following terms?
  • rotation: a transformation where a figure is turned about a fixed point
  • center of rotation: the fixed point (ex. 0,0)
  • angle of rotation: the number of degrees a figure moves clockwise or counterclockwise about the center of rotation
angle of rotation
Angle of Rotation

-the number of degrees a figure moves clockwise or counterclockwiseabout the center of rotation

angle of rotation1
Angle of Rotation

-the number of degrees a figure moves clockwise or counterclockwiseabout the center of rotation

90 clockwise rotations
90º Clockwise Rotations

Do you see a way to find the image coordinates if you have the original coordinates?

90 clockwise rotations1
90º Clockwise Rotations

Try:

Original Image

A (3, -5)

B (2, -4)

C (4, -1)

D (-1, 6)

Switch the coordinates and multiply the new y-coordinate by -1. (x,y) (y,-x)

90 clockwise rotations2
90º Clockwise Rotations

Try:

Original Image

A (3, -5) A’ (-5, -3)

B (2, -4) B’ (-4, -2)

C (4, -1) C’ (-1, -4)

D (-1, 6) D’ (6, 1)

Switch the coordinates and multiply the new y-coordinate by -1. (x,y)  (y,-x)

90 counterclockwise rotations
90º Counterclockwise Rotations

Do you see a way to find the image coordinates if you have the original coordinates?

90 counterclockwise rotations1
90º Counterclockwise Rotations

Try:

Original Image

A (3, -5)

B (2, -4)

C (4, -1)

D (-1, 6)

Switch the coordinates and multiply the new x-coordinate by -1. (x,y) (-y,x)

90 counterclockwise rotations2
90º Counterclockwise Rotations

Try:

Original Image

A (3, -5) A’ (5,3)

B (2, 4) B’ (-4,2)

C (-4, 1) C’ (-1,-4)

D (-1, 6) D’ (-6,-1)

Switch the coordinates and multiply the new x-coordinate by -1. (x,y) (-y,x)

180 rotations
180º Rotations

To rotate a figure 180º about the origin, multiply each coordinate by -1. (x, y) → (-x, -y)

rotational symmetry1

A family crest has rotational symmetry for

90º and 180º clockwise (or counterclockwise)

rotation.

Rotational Symmetry
rotational symmetry2

A family crest has rotational symmetry for

90º and 180º clockwise (or counterclockwise)

rotation.

Rotational Symmetry
rotational symmetry3

A family crest has rotational symmetry for

90º and 180º clockwise (or counterclockwise)

rotation.

Rotational Symmetry
rotational symmetry4

A family crest has rotational symmetry for

90º and 180º clockwise (or counterclockwise)

rotation.

Rotational Symmetry
dilations
Dilations

-a transformation where a figure stretches or shrinks with respect to a fixed point (center of dilation – ex. 0,0)

-in a dilation, a figure and its image are similar

what do you notice about the original and image coordinate points1
What do you notice about the original and image coordinate points?

You can multiply the coordinates by a scale factor of 2.

what do you notice about the original and image coordinate points2
What do you notice about the original and image coordinate points?

You can multiply the coordinates by a scale factor of 2.

ie. (x,y) (2x,2y)

slide33

Or the thing you are multiplying by.

If your scale factor is 2, you multiply each coordinate by 2. If it is 0.5, you multiply by 0.5

slide41
OYO

Draw quadrilateral ABCD with vertices A(-1, 2), B(1, 2), C(3, 0), and D(-1, -1). Then find the coordinates of the vertices of the image after a dilation having a scale factor of 2, and draw the image.

oyo solution
OYO Solution

Draw quadrilateral ABCD with vertices A(-1, 2), B(1, 2), C(3, 0), and D(-1, -1). Then find the coordinates of the vertices of the image after a dilation having a scale factor of 2, and draw the image.

SOLUTION

A’ (-2, 4)

B’ (2, 4)

C’ (6, 0)

D’ (-2, -2)

oyo solution cont
OYO Solution (cont.)

Draw quadrilateral ABCD with vertices A(-1, 2), B(1, 2), C(3, 0), and D(-1, -1). Then find the coordinates of the vertices of the image after a dilation having a scale factor of 2, and draw the image.

SOLUTION

A’ (-2, 4)

B’ (2, 4)

C’ (6, 0)

D’ (-2, -2)

finding a scale factor
Finding a Scale Factor

To find a scale factor, divide the larger photo coordinates by the smaller photo coordinates (since it was an enlargement).

5/2 = 2.5 2.5/1 = 2.5

12.5/5 = 2.5 2.5/1 = 2.5

The scale factor is 2.5

slide45
Try:

Find the scale factor.

Solution: A’ to A is 3 / 0.5 = 6 and B’ to B is 6/1 = 6.

The scale factor is 6. The image (the enlarged line) is 6 times larger than the original line.

homework
Homework

Green Challenge:

p.744 – 745 # 3 – 5 all, 8 – 13all, 15, 17 *3 graphs needed*

p.749 – 750 # 1, 5 – 10 all *4 graphs needed*

Blue Challenge:

p.744 – 746 # 9 – 19 (odd), 20 – 22 all *5 graphs needed*

p.749 – 751 # 1, 5 – 13all, 17, 18 *8 graphs needed*