We will now take a closer look at the ideas behind the different types of symmetries that we have discussed by studying four different rigid motions.

Transcription

1 hapter 11: The Matheatics of Syetry Sections 1-3: Rigid Motions Tuesday, pril 3, 2012 We will now take a closer look at the ideas behind the different types of syetries that we have discussed by studying four different rigid otions. Rigid Motions The act of taking an object and oving it fro soe starting position to soe ending position without altering its shape or size is called a rigid otion. Figure 1 shows an exaple of a rigid otion. Figure 2 is an exaple of a otion that is not rigid. Figure 1 Figure 2 reflection is a otion that oves an object to a irror iage of itself. The irror is called the axis of reflection, and is given by a line in the plane. Reflections We will use a capital letter, such as F, to denote the original object, and, in this case, F to denote the iage of the object under the transforation. To find the iage of a point P under a reflection, draw the line through P that is perpendicular to the axis of reflection. The iage P will be the point on this line whose distance fro is the sae as that between P and. P P Exaple 5: Find the iages of P and Q under their respectively indicated reflections. P Q Q P

2 Exaple 6: Find the iage of the quadrilateral under the reflection given by the line. Exaple 7: Find the iage of the triangle under the reflection given by the line. Exaple 8: Find the axis of reflection,, for the reflection that takes P to P. P P

3 Properties of Reflections Property 1: reflection is copletely deterined by its axis of reflection. Property2: reflection is copletely deterined by a single point-iage pair P and P (if P P ). Property 3: reflection has infinitely any fixed points (all points on the line ). Reeber, a fixed point of a otion is a point that is oved onto itself. Property 4: reflection is an iproper otion because it reverses the orientation of objects. Property 5: pplying the sae reflection twice is equivalent to not oving the object at all. So applying a reflection twice results in the identity otion. Rotations rotation is a otion that swings an object around a fixed point. The fixed center point of the rotation is called the rotocenter. The aount of swing is given by the angle of rotation. n exaple of a clockwise rotation with rotocenter O and angle of rotation α. It oves the point P to the point P.

4 Exaple 9: Find the iage of triangle under a 90 counterclockwise rotation with rotocenter O. O Notice that the distance of each point fro the rotocenter O does not change under the rotation. s a convention, any angle in the counterclockwise direction has a positive angle easure. ny angle in the clockwise direction has a negative angle easure. F E Exaple 10: Find the iage of under the rotation with rotocenter O and angle of rotation O

5 Exaple 11: Find the iage of under the rotation with rotocenter O and angle of rotation 45. E E O Properties of Rotations Property 1: rotation is copletely deterined by two point-iage pairs. Property 2: rotation has one fixed point, the rotocenter. Property 3: rotation is a proper otion - the orientation of the object is aintained. Property 4: 360 rotation around any rotocenter is equivalent to the identity otion.

6 Exaple 12: In each case, give an answer between 0 and 360. clockwise rotation by an angle of 710 is equivalent to a clockwise rotation by an angle of = = 350 clockwise rotation by an angle of 710 is equivalent to a counterclockwise rotation by an angle of... clockwise rotation of 710 is equivalent to a clockwise rotation of 350 which is equivalent to a counterclockwise rotation of = 10 clockwise rotation by an angle of is equivalent to a clockwise rotation by an angle of = (360 ) = 80 clockwise rotation by an angle of is equivalent to a counterclockwise rotation by an angle of... clockwise rotation of is equivalent to a clockwise rotation of 80 which is equivalent to a counterclockwise rotation of = 280 Exaple 13: rotation oves to and to. Find the rotocenter and the iage of under the rotation. O