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Lesson 10-5

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Transformations

Lesson 10-5: 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.

Lesson 10-5: Transformations

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

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

Example:

The figure is reflected across line l .

l

Lesson 10-5: Transformations

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

- reflects across the y axis to line n
(2, 1) (-2, 1) & (5, 4) (-5, 4)

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 :

n

l

- reflects across the x axis to line m.
(2, 1) (2, -1) & (5, 4) (5, -4)

m

Lesson 10-5: Transformations

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.

Example:

Reflect the fig. across the line y = 1.

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

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

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

Lesson 10-5: Transformations

- 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.

Four lines of symmetry

One line of symmetry

Two lines of symmetry

Infinite lines of symmetry

No lines of symmetry

Lesson 10-5: Transformations

- 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).

Example:

A

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

B

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

Lesson 10-5: Transformations

- 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.

Example:

C

B

A

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

Lesson 10-5: Transformations

A

C

m

B

n

- 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.

Lesson 10-5: Transformations

It 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)

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

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)

Lesson 10-5: Transformations

B

A

35 °

- 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.

Lesson 10-5: Transformations

K

B

A

R

X

35 °

- 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.

Lesson 10-5: Transformations

- 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.)

Lesson 10-5: Transformations

E

A

F

C

B

A

R

B

C

W

- 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.

Lesson 10-5: Transformations

Find the measure of the dilation image of segment AB, 6 units long, with a scale factor of

- 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.
- 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.
- 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.

Lesson 10-5: Transformations