DRILL 1) If A is in between points B and C and AC is 4x + 12 and AB is 3x – 4 and BC is 57 feet how long is AB? 2) Angles A and B are Supplementary if Angle A is 2x – 20 and Angle B is 4x – 44. Find the measure of Angle A.

3.1 thru 3.3 Motions in Geometry Geometry Chapter 3 Transformations:

3.1 Reflections

Transformations  A transformation is any type of movement in geometry, it can be a change in shape, size, or simply location of an object.  The three types of Transformations we will talk about today are Reflections, Rotations and Translations.

Pre-Image And Image  Pre-Image is the original figure before any type of transformation takes place.  Image is the new figure after the transformation has taken place.

Vocabulary Isometry: a transformation in which the original figure and it’s image are congruent. Isometry: a transformation in which the original figure and it’s image are congruent. Opposite Orientation: when an image appears to be backwards compared to the pre-image. Opposite Orientation: when an image appears to be backwards compared to the pre-image.

Reflection  A transformation in which a line of reflection acts as a mirror reflecting points from their pre-image to their image.

Reflections  A reflection reverses orientation.  A reflection is an isometry.  A reflection over the x-axis results in a change in the y-coordinate.  A reflection in the y-axis results in a change in the x-coordinate.

Reflections in Coordinate Plane  When reflecting a point over the x- axis the y-coordinate changes sign. (x, y)  (x, -y)  When reflecting over the y-axis the x-coordinate changes sign. (x, y)  (-x, y)  When reflecting over the origin both the x and y coordinates change signs. (x, y)  (-x, -y)

Examples 1) If you reflect the point (6, -1) over the y-axis what would your new point be? 2) If you reflect the point (-2, 3) over the y-axis what would your new point be? 3) Translate (2, 4) up 3 and left 1. What is your new point?

12.2 Translations

Translations  A translation is a sliding of a figure from one point to another. Since a sliding of a figure would not change the figures shape or size it is known as a Rigid Motion.

Vocabulary Translation: is a transformation where you are sliding the object without changing orientation. Translation: is a transformation where you are sliding the object without changing orientation. * A translation is an isometry * A translation is an isometry Composition: is when two transformations are performed one right after the other. Composition: is when two transformations are performed one right after the other.

Examples of Translation  To perform a translation simply add or subtract from the coordinates of each point on the figure.  If we want to translate the point (4, 6) up 4 and left 3. We would simply add 4 to the “y” and subtract 3 from the “x”. We would get the new point (1, 10).

Translation  To translate a point in a coordinate plane simply add or subtract to the x or y coordinates.  To move the point (2, 4) up 3 units you would have to add 3 to the y-coordinate (4). So (2, 4) would become ( 2, 4 + 3) or (2, 7)

Vector Notation  Vector notation is used to show what you are doing to each coordinate to get your new coordinates.  The vector mean you subtract 3 from the x-coordinates and add 5 to the y- coordinates in order to get your new points.

3.3 Rotations

Rotation  A rigid motion that moves a geometric figure about a point known as the turn center.

Properties of a Rotation  A rotation is an Isometry.  A rotation does not change orientation.

Finding The Angle Of Rotation  Find the number of congruent images under a rotation and then divide that number into 360.  EX:

Rotation of 180 Degrees  A Rotation of 180 Degrees is equivalent to a reflection over the origin.  (x, y) becomes (-x, -y)