Math 10F

Transformational Geometry Examples

Translations• Translations are “slides”• Described by a length and

direction• Eg. translate the

following shape 6units left, and 6 units down…

Translations• 6 units left…

Translations• 6 units left…• 6 units down

Translations• 6 units left…• 6 units down• Do the same

translation for each key point

Translations• Another example: Translate this

shape 5 units left, and 3 units up

Translations• 5 left and 3 up• One point

Translations• 5 left and 3 up• Two Points

Translations• 5 left and 3 up• 3 Points

Translations• 5 left and 3 up• All Points (Connect)

Translations• Mapping Notation• (x,y) (x+2,y- 4)

Translations• Mapping Notation• (x,y) (x+2,y- 4)• This means

“right 2”, “down 4”

Translations• Mapping Notation• (x,y) (x+2,y- 4)• This means

“right 2”, “down 4”• All 4 key points

Translations• Mapping Notation• (x,y) (x+2,y- 4)• This means

“right 2”, “down 4”• All 4 key points• Connect

Translations• Show the following translations:• (x,y) (x+2, y+6)• Up 4, left 3• [-4,-1] (ordered

pair notation)

Translations• Show the following translations:• (x,y) (x+2, y+6)• Up 4, left 3• [-4,-1] (ordered

pair notation)

Translations• Show the following translations:• (x,y) (x+2, y+6)• Up 4, left 3• [-4,-1] (ordered

pair notation)

Translations• Show the following translations:• (x,y) (x+2, y+6)• Up 4, left 3• [-4,-1] (ordered

pair notation)

Reflections

• Reflections are transformations in which a figure is reflected or flipped over a reflection line.

• Each point in the new figure is the same perpendicular distance from the reflection line as the old point, except on the other side of the line.

Reflections• Reflect the shape through the y-

axis.

Reflections• Reflect the shape through the y-

axis.• Find the

perpendiculardistances of thekey points.

Reflections• Reflect the shape through the y-

axis.• Find the

perpendiculardistances of thekey points.

• Find correspondingdistanceson the other side.

Reflections• Reflect the shape through the y-

axis.• Find the

perpendiculardistances of thekey points.

• Find correspondingdistanceson the other side.

• Connect the points

Reflections• Reflect the shape through the x-

axis.

Reflections• Reflect the shape through the x-

axis.• Find the

perpendiculardistances of thekey points.

Reflections• Reflect the shape through the x-

axis.• Find the

perpendiculardistances of thekey points.

• Find correspondingdistanceson the other side.

Reflections• Reflect the shape through the x-

axis.• Find the

perpendiculardistances of thekey points.

• Find correspondingdistanceson the other side.

• Connect the points

Reflections• Reflect the shape through the

given reflection line.

Reflections• Reflect the shape through the

given reflection line.• Find the

perpendiculardistances of thekey points.

Reflections• Reflect the shape through the

given reflection line.• Find the

perpendiculardistances of thekey points.

• Find correspondingdistanceson the other side.

Reflections• Reflect the shape through the

given reflection line.• Find the

perpendiculardistances of thekey points.

• Find correspondingdistanceson the other side.

• Connect the points

Rotations• A rotation is a transformation in

which a figure is turned or rotated about a point.

• Rotations areCW or CCW

• All rotations turnrelative to thecentre of rotation.

• All rotations in thiscourse will be in90º increments.

Rotations• Rotate the figure 90º CCW through

the origin

Rotations• Rotate the figure 90º CCW through

the origin• Pick a key point• Measure a horizontal

distance and a vertical distanceto the turn centre

Rotations• Rotate the figure 90º CCW through the

origin• Pick a key point• Measure a horizontal

distance and a vertical distanceto the turn centre

• The horizontaldistance becomesyour new vertical,and your old verticalbecomes your new horizontal.

Rotations• Rotate the figure 90º CCW through

the origin

• Do this for EACHkey point

Rotations• Rotate the figure 90º CCW through

the origin

• Do this for EACHkey point

Rotations• Rotate the figure 90º CCW through

the origin

• Connect the newpoints to showthe rotated figure

Rotations• Rotate the figure 180º CW through

the origin

Rotations• Rotate the figure 180º CW through

the origin

Rotations• Rotate the figure 90º CW through

(0,2)

Rotations• Rotate the figure 90º CW through

(0,2)

Rotations• Rotate the figure 90º CW through

(3,3)

Rotations• Rotate the figure 90º CW through

(3,3)

Dilations• Dilations are enlargements or

reductions of a figure• The dilation is

enlarged by the scale factor(a multiplier)

• In this class, alldilations will occurabout the origin.

Dilations• Dilate the following figure by a

scale factor of 3

Dilations• Dilate the following figure by a

scale factor of 3

Dilations• Dilate the following figure by a

scale factor of 1/2

Dilations• Dilate the following figure by a

scale factor of ½• Notice that for

dilations, if your scale factor is >1,you are magnifying,and if your SF <1,you are shrinking

Dilations• Dilations can be expressed using a

mapping notation• Eg. perform the

following dilationon the figure given(x,y)(2x,2y)

Dilations• Dilations can be expressed using a

mapping notation• Eg. perform the

following dilationon the figure given(x,y)(2x,2y)

Dilations• For your dilations, the distances

from the origin for all key points will be proportional betweenyour two figures