DRILL 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? Angles A and B are Supplementary if Angle A is 2x – 20 and Angle B is 4x – 44. Find the measure of Angle A.
Unit 2 Transformations: Honors Geometry
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.
Reflections
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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. 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 If you reflect the point (6, -1) over the y-axis what would your new point be? If you reflect the point (-2, 3) over the x-axis what would your new point be? Reflect (-2, 4) over the origin. What is your new point? Answers (-6, -1) (-2, -3) (2, -4)
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. * A translation is an isometry 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.
Vocabulary Glide Reflection: a glide reflection is simply when you translate a figure as well as reflect it over a line.
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: The image has 4 congruent views, so 360/4 = 90. The image has a 90o angle of rotation.
Rotation of 180 Degrees A Rotation of 180 Degrees about the origin, is equivalent to a reflection over the origin. (x, y) becomes (-x, -y)
Examples What would be the new point formed when you reflect the point (-2, 8) over the origin? If you translate the point (-1, -4) using the vector , what would be the new point? If the coordinates of A are (4, -2) and the coordinates of are (-2, 3) what vector was used to get the new point?