Transformations

Lesson 10-5: Transformations 1

Transformations


Types of Transformations

Reflections: These are like mirror images as seen across a line or a point.

Translations ( or slides): This moves the figure to a new location with no change to the looks of the figure.

Rotations: This turns the figure clockwise or counter-clockwise but doesn’t change the figure.

Dilations: This reduces or enlarges the figure to a similar figure.


Reflections

You could fold the picture along line l and the left figure would coincide with the corresponding parts of right figure.

l

You can reflect a figure using a line or a point. All measures (lines and angles) are preserved but in a mirror image.

Example: The figure is reflected across line l .


Reflections – continued…

reflects across the y axis to line n

(2, 1) (-2, 1) & (5, 4) (-5, 4)

Reflection across the x-axis: the x values stay the same and the y values change sign. (x , y) (x, -y)

Reflection across the y-axis: the y values stay the same and the x values change sign. (x , y) (-x, y)

Example: In this figure, line l :

reflects across the x axis to line m.

(2, 1) (2, -1) & (5, 4) (5, -4)

ln

m


Reflections across specific lines:

To reflect a figure across the line y = a or x = a, mark the corresponding points equidistant from the line.

i.e. If a point is 2 units above the line its corresponding image point must be 2 points below the line.

(-3, 6) (-3, -4)

(-6, 2) (-6, 0)

(2, 3) (2, -1).

Example:

Reflect the fig. across the line y = 1.


Lines of Symmetry If a line can be drawn through a figure so the one side of the

figure is a reflection of the other side, the line is called a “line of symmetry.”

Some figures have 1 or more lines of symmetry. Some have no lines of symmetry.

One line of symmetry

Infinite lines of symmetry

Four lines of symmetry

Two lines of symmetry

No lines of symmetry


Translations (slides) If a figure is simply moved to another location without change to

its shape or direction, it is called a translation (or slide). If a point is moved “a” units to the right and “b” units up, then

the translated point will be at (x + a, y + b). If a point is moved “a” units to the left and “b” units down, then

the translated point will be at (x - a, y - b).

A

B

Image A translates to image B by moving to the right 3 units and down 8 units.

Example:

A (2, 5) B (2+3, 5-8) B (5, -3)


Composite Reflections

If an image is reflected over a line and then that image is reflected over a parallel line (called a composite reflection), it results in a translation.

A B C

Image A reflects to image B, which then reflects to image C. Image C is a translation of image A

Example:


Rotations

An image can be rotated about a fixed point.The blades of a fan rotate about a fixed point.

An image can be rotated over two intersecting lines by using composite reflections.

Image A reflects over line m to B, image B reflects over line n to C. Image C is a rotation of image A.

A

B

C

mn


RotationsIt is a type of transformation where the object is rotated around a fixed

point called the point of rotation.

When a figure is rotated 90° counterclockwise about the origin, switch each coordinate and multiply the first coordinate by -1. (x, y) (-y, x)

When a figure is rotated 180° about the origin, multiply both coordinates by -1.

(x, y) (-x, -y)

Ex: (1,2) (-1,-2) & (6,2) (-6, -2)

Ex: (1,2) (-2,1) & (6,2) (-2, 6)


Angles of rotation In a given rotation, where A is the figure and B is the resulting

figure after rotation, and X is the center of the rotation, the measure of the angle of rotation AXB is twice the measure of the angle formed by the intersecting lines of reflection.

Example: Given segment AB to be rotated over lines l and m, which intersect to form a 35° angle. Find the rotation image segment KR.

A

B

35 °


Angles of Rotation . . Since the angle formed by the lines is 35°, the angle of rotation is

70°. 1. Draw AXK so that its measure is 70° and AX = XK. 2. Draw BXR to measure 70° and BX = XR. 3. Connect K to R to form the rotation image of segment AB.

A

B

35 °X

K

R


Dilations A dilation is a transformation which changes the size of a figure

but not its shape. This is called a similarity transformation. Since a dilation changes figures proportionately, it has a scale

factor k.• If the absolute value of k is greater than 1, the dilation is an

enlargement.• If the absolute value of k is between 0 and 1, the dilation is a

reduction. • If the absolute value of k is equal to 0, the dilation is

congruence transformation. (No size change occurs.)


Dilations – continued… In the figure, the center is C. The distance from C to E is three times the distance

from C to A. The distance from C to F is three times the distance from C to B. This shows a transformation of segment AB with center C and a scale factor of 3 to the enlarged segment EF.

In this figure, the distance from C to R is ½ the distance from C to A. The distance from C to W is ½ the distance from C to B. This is a transformation of segment AB with center C and a scale factor of ½ to the reduced segment RW.

C

E

A

FB

C

R

A

BW


Dilations – examples…Find the measure of the dilation image of segment AB, 6 units long,

with a scale factor of 1. S.F. = -4: the dilation image will be an enlargment since the

absolute value of the scale factor is greater than 1. The image will be 24 units long.

2. S.F. = 2/3: since the scale factor is between 0 and 1, the image will be a reduction. The image will be 2/3 times 6 or 4 units long.

3. S.F. = 1: since the scale factor is 1, this will be a congruence transformation. The image will be the same length as the original segment, 1 unit long.

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Transformations