Rotations and Dilations

ROTATION

ROTATION

A rotation is a transformation that turns a figure about (around) a point or a line.

The point a figure turns around is called the center of rotation.

Basically, rotation means to spin a shape.

The center of rotation can be on, outside or even inside the shape.

ROTATION

What does a rotation look like?

A ROTATION MEANS TO TURN A FIGURE

A ROTATION MEANS TO TURN A FIGURE

center of rotation

ROTATION

This is another way rotation looks.

A ROTATION MEANS TO TURN A FIGUREA ROTATION MEANS TO TURN A FIGURE

center of rotation

The triangle was rotated around the

point.

ROTATIONDescribe how the triangle A was transformed to

make triangle B

A B

Describe the translation.Triangle A was rotated right 90Triangle A was rotated right 90

ROTATIONDescribe how the arrow A was transformed to

make arrow B

Describe the translation.Arrow A was rotated 180 Arrow A was rotated 180

A

B

Rotations with the Origin as the Center

90o Clockwise Rotation About the Origin:

( , ) ( , )x y y x

90o Counterclockwise Rotation about the Origin

( , ) ( , )x y y x

180o Rotation Clockwise or Counterclockwise about the Origin

( , ) ( , )x y x y

Rotations by Another Name

A 270o clockwise rotation about the origin is the same effect as a 90o counterclockwise rotation about the origin.

A 270o counterclockwise rotation about the origin is the same effect as a 90o clockwise rotation about the origin.

( , ) ( , )x y y x

( , ) ( , )x y y x

ROTATION

When some shapes are rotated they create a special situation called

rotational symmetry.

to spin a shape the exact same

Most playing cards have a rotational symmetry of 180o so you don’t have to turn your cards the right way.

ROTATIONAL SYMMETRYA shape has rotational symmetry if, after you rotate 180o or less, it is the

same as the original shape.Here is an example…

As this shape is rotated 360, is it ever the same before the shape returns to its original direction?

Yes, when it is rotated 90 it is the same as it was in the beginning.

So this shape is said to have rotational symmetry.

90

ROTATIONAL SYMMETRY

Here is another example…

As this shape is rotated 360, is it ever the same before the shape returns to its original direction?

Yes, when it is rotated 180 it is the same as it was in the beginning.

So this shape is said to have rotational symmetry.180

A shape has rotational symmetry if, after you rotate one half-turn or less, it is the

same as the original shape.

ROTATIONAL SYMMETRY

Here is another example…

As this shape is rotated 360, is it ever the same before the shape returns to its original direction?

No, when it is rotated 360 it is never the same.

So this shape does NOT have rotational symmetry.

A shape has rotational symmetry if, after you rotate one half-turn or less, it is the

same as the original shape.

ROTATIONAL SYMMETRY

Does this shape have rotational symmetry?

Yes, when the shape is rotated 60o , 120

and 180o it is the same. Since these

are all less than 180, this shape HAS

rotational symmetry.

Dilation changes the size of the figure without changing the shape.

DILATION

When you enlarge a photograph or use a copy machine to reduce a map, you are making dilations.

An Enlargement means the shape is bigger and the scale factor is greater than 1.

DILATION

A Reduction means the new shape is smaller and the scale factor is between 0 and 1.

The scale factor tells you by what factor something is enlarged or reduced.

DILATIONNotice each time the rabbit transforms the shape stays

the same and only the size changes.

Look at the pictures below

DILATION

Dilate the image with a scale factor of 75%

Dilate the image with a scale factor of 150%

Look at the pictures below

DILATION

Dilate the image with a scale factor of 100%

Why is a dilation of 75% smaller, a dilation of 150% bigger, and a dilation of 100% the

same?

The Scale Factor of a Dilation Centered at Point C

If C and P are distinct points, you can find the scale factor of a dilation centered at C by the following equation.

'CPScale factor

CP

Dilations Centered at the Origin

If a dilation is centered at the origin, which is often the case, you can use the scale factor to easily find the image coordinates. All you have to do is multiply the pre-image coordinates by the scale factor. You can also find the coordinates of the pre-image by dividing the image coordinates by the scale factor. Do remember, this only works for dilations centered at the origin.

Example:Find the coordinates of the image of when dilated by a scale factor of 3 centered at the origin.

(3, 3), (4,2) ( 2,5)ABC with A B and C

Solution:Multiply all x- and y-coordinates by 3. (3, 3) '(9, 9)

(4,2) '(12,6)

( 2,5) '( 6,15)

A A

B B

C C

SummaryRotations and dilations are transformations. Rotations do not change the size or shape of a figure, they simply turn them about a fixed point. Dilations do change the size of a figure but not the shape. There are rules that make it easier to find image points of both of these transformations when they are centered at the origin.

This book is being rotated and dilated!