CK-12 Geometry Concepts

12.4. Rotations http://www.ck12.org

1. Draw 4 ABCand a pointRoutside the circle.


2. Draw the line segmentRB.


3. Take your protractor, place the center onRand the initial side onRB. Mark a 100◦angle.


4. FindB′such thatRB=RB′.


5. Repeat steps 2-4 with pointsAandC.


6. ConnectA′,B′,andC′to form 4 A′B′C′.

This is the process you would follow to rotate any figure 100◦counterclockwise. If it was a different angle measure,
then in Step 3, you would mark a different angle. You will need to repeat steps 2-4 for every vertex of the shape.

Common Rotations

• Rotation of 180 ◦:If(x,y)is rotated 180◦around the origin, then the image will be(−x,−y).


• Rotation of 90 ◦:If(x,y)is rotated 90◦around the origin, then the image will be(−y,x).


• Rotation of 270 ◦:If(x,y)is rotated 270◦around the origin, then the image will be(y,−x).

While we can rotate any image any amount of degrees, only 90◦, 180 ◦and 270◦have special rules. To rotate a figure
by an angle measure other than these three, you must use the process from the Investigation.

Example A

Rotate 4 ABC, with verticesA( 7 , 4 ),B( 6 , 1 ),andC( 3 , 1 ) 180 ◦. Find the coordinates of 4 A′B′C′.

It is very helpful to graph the triangle. IfAis( 7 , 4 ), that means it is 7 units to the right of the origin and 4 units up.
A′would then be 7 units to theleftof the origin and 4 unitsdown.The vertices are:

A( 7 , 4 )→A′(− 7 ,− 4 )

B( 6 , 1 )→B′(− 6 ,− 1 )

C( 3 , 1 )→C′(− 3 ,− 1 )

Example B

RotateST 90 ◦counter-clockwise about the origin.

Using the 90◦rotation rule,T′is (8, 2).

Example C

Find the coordinates ofABCDafter a 270◦rotation counter-clockwise about the origin.

Using the rule, we have:

(x,y)→(y,−x)

A(− 4 , 5 )→A′( 5 , 4 )

B( 1 , 2 )→B′( 2 ,− 1 )

C(− 6 ,− 2 )→C′(− 2 , 6 )

D(− 8 , 3 )→D′( 3 , 8 )