Chapter 7. Transformations. Chapter Objectives. Identify different types of transformations Define isometry Identify reflection and its characteristics Identify rotations and the characteristics Identify translations and the characteristics
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Chapter 7 Transformations
Chapter Objectives • Identify different types of transformations • Define isometry • Identify reflection and its characteristics • Identify rotations and the characteristics • Identify translations and the characteristics • Identify a glide reflection and its characteristics • Describe composition transformations
Lesson 7.1 Rigid Motion in a Plane
Definition of Transformation • A transformation is any operation that maps, or moves, an object to another location or orientation.
Transformation Terms • When performing a transformation, the original figureis called the pre-image. • The new figure is called the image. • Many transformations involve labels typically using letters of the alphabet. • The imageis named after the pre-image, by adding a prime symbol (apostrophe) AA’A’’ • We say it as “A prime”
Types of Transformations • There are 3 basic transformations: • A reflection in a line. • A rotations about a point. • A translation.
Example 1 • Identify the following transformations: • Rotation • Reflection • Translation
Isometry • An isometry is a transformation that preserves the following: • length • angle measures • parallel lines • distance between points • An isometry is also called a rigid transformation.
ST WV Example 2 Use the figure shown where QRST is mapped onto figure VWXY: • Name the preimage of XY • Name the image of QR • Name a triangle that appears to be congruent to WXY RST R W Q S X V T Y
Lesson 7.2 Reflections
Reflections • A transformation that uses a line like a mirror is called a reflection. • The line that acts like a mirror is called the line of reflection. • When you talk of a reflection, you must include your line of reflection • A reflection in a line m is a transformation that maps every point P in the plane to a point P’, so that the following is true • If P is not on line m, then m is the perpendicular bisector of PP’. • If P is on line m, then P = P’.
Example 3 Give the image of the following reflections: • K(3,5) in the y-axis • K’(-3,5) • W(2,-7) in the line y = -2 • W’(2,3)
Reflection Formula • There is a formula to all reflections. • It depends on which type of a line are you reflecting in. • vertical • horizontal • y = x Vertical: y-axis Horizontal: x-axis y = x ( x , y) ( x , -y) ( -x, y) y = a x = a ( y , x) ( x , -y + 2a) ( -x + 2a, y)
m Theorem 7.1:Reflection Theorem • A reflection is an isometry. • That means a reflection does not change the shape or size of an object!
Line of Symmetry • A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in a line. • What that means is a line can be drawn through an object so that each side reflects onto itself. • There can be more than one line of symmetry, in fact a circle has infinitely many around.
6 2 1 4 5 3 Example 4 Determine the number of lines of symmetry for the following figures:
Minimum Distance • Imagine you are building a house and you wish to place one power outlet in the kitchen so it will run the microwave, M, and the can opener, C. In order to place the outlet such that it will use the minimum amount of cord for each instrument you should do the following: • Reflect M in the line created by the wall. • Then draw a line connecting M’ and C. • The point at which line M’C intersects the line created by the wall is where the location of the outlet should be to create the minimum distance used for the power cords.
Lesson 7.3 Rotations
Q Definitions of Rotations • A rotation is a transformation in which a figure is turned about a fixed point. • The fixed point is called the center of rotation. • The amount that the object is turned is the angle of rotation. • The rotation with either be in a clockwise direction or a counterclockwise direction. 90 oclockwise
A A’ B B’ P Theorem 7.2:Rotation Theorem • A rotation is an isometry.
Rotating About the Origin • Rotating about the origin in 90o turns is like reflecting in the line y = x and in an axis at the same time! • So that means to switch the positions of x and y. • (x,y) (y,x) • Then the original x-value, now new y-value, changes sign, no matter where it is flipped to. • So overall the transformation can be described by • (x,y) (-y,x) • Every time you 90o you repeat the process. • So going 180o means you do the process twice!
Example 5 Graph quadrilateral RSTU with the following vertices and then rotate the quadrilateral 270o counterclockwise about the origin.
Rotational Symmetry • A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less. • A square has rotational symmetry because it maps onto itself with a 90° rotation, which is less than 180°. • A rectangle has rotational symmetry because it maps onto itself with a 180° rotation. 90o 90o 45o 45o
Example 6 Determine whether the figure has rotational symmetry. If so, describe the rotations that would map the figure onto itself. • 45o, 90o, 135o, 180o • 180o • None
A m k B 2x x A’ P B’ Theorem 7.3:Angle of Rotation Theorem • The angle of rotation is twice as big as the angle of intersection. • But the intersection must be the center of rotation. • And the angle of intersection must be acute or right only.
Lesson 7.4 Translations and Vectors
Translation Definition • A translation is a transformation that maps an object by shifting or sliding the object and all of its parts in a straight line. • A translation must also move the entire object the same distance.
Theorem 7.4:Translation Theorem • A translation is an isometry.
Q’ Q’’ Q P’ P’’ P k m Theorem 7.5:Distance of Translation Theorem • If lines k and m are parallel, then a reflection in line k followed by a reflection in the line m can also be mapped as a translation. If P’’ is the image of P, then the following is true: • PP’’ is perpendicular to line k and line m. • The distance of the translation is twice the distance between the reflecting lines. x 2x
Coordinate form • Every translation has a horizontal movement and a vertical movement. • A translation can be described in coordinate notation. • (x,y) (x+a , y+b) • Which tells you to move a units horizontal and b units vertical. Q b units up P a units to the right
Example 7 Find the image of each vertex of quadrilateral ABCD after the translation (x,y) (x+3,y-1)
Vectors • Another way to describe a translation is to use a vector. • A vectoris a quantity that shows both direction and magnitude, or size. • It is represented by an arrow pointing from pre-image to image. • The starting point at the pre-image is called the initial point. • The ending point at the imageis called the terminal point.
Component Form of Vectors • Component form of a vector is a way of combining the individual movements of a vector into a more simple form. • <x , y> • Naming a vector is the same as naming a ray. • PQ • Vectors work the same as coordinate form • <x , y> = (a,b) (a+x,b+y) Q y units up P x units to the right
Example 8 Name the vector and write it in component form. • CD = <7,-3> • JP = <0,4>
Lesson 7.5 Glide Reflections and Compositions
Glide Reflection Definition • A glide reflection is a transformation in which a reflection and a translation are performed one after another. • The translationmust be parallel to the line of reflection. • As long as this is true, then the order in which the transformation is performed does not matter!
Compositions of Transformations • When two or more transformations are combined to produce a single transformation, the result is called a composition. • So a glide reflection is a composition. • The order of compositions is important! • A rotation 90o CCW followed by a reflection in the y-axis yields a different result when performed in a different order.
Theorem 7.6:Composition Theorem • The composition of two (or more) isometries is an isometry.