Math 10F Transformational Geometry Examples. Translations Translations are “slides” Described by a length and direction Eg. translate the following shape.

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Presentation transcript:

Math 10F Transformational Geometry Examples

Translations Translations are “slides” Described by a length and direction Eg. translate the following shape 6 units 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 perpendicular distances of the key points.

Reflections Reflect the shape through the y-axis. Find the perpendicular distances of the key points. Find corresponding distances on the other side.

Reflections Reflect the shape through the y-axis. Find the perpendicular distances of the key points. Find corresponding distances on the other side. Connect the points

Reflections Reflect the shape through the x-axis.

Reflections Reflect the shape through the x-axis. Find the perpendicular distances of the key points.

Reflections Reflect the shape through the x-axis. Find the perpendicular distances of the key points. Find corresponding distances on the other side.

Reflections Reflect the shape through the x-axis. Find the perpendicular distances of the key points. Find corresponding distances on 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 perpendicular distances of the key points.

Reflections Reflect the shape through the given reflection line. Find the perpendicular distances of the key points. Find corresponding distances on the other side.

Reflections Reflect the shape through the given reflection line. Find the perpendicular distances of the key points. Find corresponding distances on the other side. Connect the points

Rotations A rotation is a transformation in which a figure is turned or rotated about a point. Rotations are CW or CCW All rotations turn relative to the centre of rotation. All rotations in this course will be in 90º 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 distance to the turn centre

Rotations Rotate the figure 90º CCW through the origin Pick a key point Measure a horizontal distance and a vertical distance to the turn centre The horizontal distance becomes your new vertical, and your old vertical becomes your new horizontal.

Rotations Rotate the figure 90º CCW through the origin Do this for EACH key point

Rotations Rotate the figure 90º CCW through the origin Do this for EACH key point

Rotations Rotate the figure 90º CCW through the origin Connect the new points to show the 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, all dilations will occur about 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 dilation on the figure given (x,y)  (2x,2y)

Dilations Dilations can be expressed using a mapping notation Eg. perform the following dilation on the figure given (x,y)  (2x,2y)

Dilations For your dilations, the distances from the origin for all key points will be proportional between your two figures