Rotations

Advanced Geometry

Rigid Transformations

Lesson 3

all points of a figure

a specified _______ and __________ about

a fixed point.

Turn

Rotation

angle direction

http://www.mathsisfun.com/flash.php?path=%2Fgeometry/images/rotation.swf&w=670.5&h=571.5&col=%23FFFFFF&title=Geometry+Rotation

To rotate a figure by hand you need:• a pencil• a straightedge• a compass• a protractor

Is a rotation by hand a construction? Explain.

No. A protractor is used to measure angles. Since measurements are being used, this is

not a construction.

Draw a RotationExample:Rotate quadrilateral ABCD 45° counterclockwise about point X. • Draw a segment from X to A.

• Measure a 45° angle with as a side. Draw the other side.

XA

• Use the compass to copy onto the new ray.

XA

• Repeat this process for points B, C, and D.• Connect points A', B', C', and D'.

Example:Triangle LMN has vertices L(-2, -1), M(-1, 2), and N(1, -1). Draw the image of ∆LMN under a rotation of 115° clockwise about the point Y(-4, -2).

Common Rotations about the Origin

To rotate a figure 90º counterclockwise, about the origin take each vertex and:

1. switch the coordinates

2. multiply the first coordinate (the new x) by -1

F(2, 2)

G(4, 1.5)

H(5, 4)

F (-2, 2)

G (-1.5, 4)H(-4, 5)

To rotate a figure 180º, multiply both coordinates by -1.

R(-3, 4)

S(0, 5)

T(0, 1)

R (3, -4)

S (0, -4)

T(0, -1)

Q(-3, 2) Q (3, -2)

Example: Triangle XYZ has vertices X(-4, 1), Y(-1, 5), and Z(-6, 9). Find the coordinates of the vertices after a 90º counterclockwise rotation and graph the image.

Example: Rectangle ABCD has vertices A(-1, 1), B(-5, 1), C(-5, 4), and D(-1, 4). Find the coordinates of the vertices after a 180º counterclockwise rotation and graph the image.

Rotational SymmetryTurn the figure LESS THAN 360° so the image is the same as the pre-image.

Order Magnitudethe number of

rotations less than 360° (including 0 °)

the measure of each angle of rotation

360

order

5 72°