Transcript 9.3 Rotations 9.5 Dilations
Advanced Geometry Rigid Transformations Lesson 3
Rotations
Rotation
Turn
all points of a figure a fixed point.
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 Rotation
Example:
Rotate quadrilateral ABCD 45 ° counterclockwise about point X.
• Draw a segment from X to A.
• Measure a 45° angle with
XA
as a side. Draw the other side.
• Use the compass to copy
XA
onto the new ray. • 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)
Q(-3, 2) R(-3, 4) S(0, 5) T(0, 1) Q (3, -2) R (3, -4) S (0, -4) T (0, -1) To rotate a figure 180º, multiply both coordinates by -1.
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 Symmetry
Turn the figure LESS THAN 360 ° so the image is the same as the pre-image.
Order
the number of rotations less than 360 ° (including 0 °)
Magnitude
the measure of each angle of rotation 360
order
5 72 °