Chapter 9 Transformations
4.8 Transformations • An operation that moves or changes a geometric figure (a preimage) in some way to produce a new figure (an image).
Congruence transformations • changes the position of the figure without changing the size or shape.
A Translation • moves every point of a figure the same distance in the same direction. • Coordinate notation: (x , y) (x + a, y + b)
Example • The vertices of ABCare A(4, 4), B(6, 6), and C(7, 4). The notation (x, y)→ (x + 1, y – 3) describes the translation of ABC to DEF. • What are the vertices of DEF?
A Reflection • Uses a line of reflection to create a mirror image of the original figure. • Coordinate notation for reflection in the x-axis : (x ,y) (x , -y) • Coordinate notation for reflection in the y- axis: (x , y) (-x, y)
Example • Reflect a figure in the x-axis
Rotation • Turns a figure about a fixed point called the center of rotation
Examples • Graph ABand CD.Tell whether CDis a rotation of ABabout the origin. If so, give the angle and direction of rotation. • A(–3, 1),B(–1, 3),C(1, 3),D(3, 1)
Tell whether PQRis a rotation of STR. If so, give the angle and direction of rotation.
6.7 Dilations • A transformation that stretches or shrinks a figure to create a similar figure. • A figure is reduced or enlarged with respect to a fixed point called the center of dilation.
The scale factor of a dilation is the ratio of the side length of the image to the corresponding side length of the original figure • Coordinate notation for a dilation with respect to the origin: (x ,y) ( kx, ky) • Reduction: 0 < k < 1 • Enlargement : k > 1
Examples • Draw a dilation of quadrilateralABCDwith verticesA(2, 1), B(4, 1), C(4, – 1), andD(1, – 1). Use a scale factor of2.
9.1 Translating Figures and Using Vectors • Translation Theorem: A translation is an isometry. • Isometry- a congruence transformation • Preimage- original figure • Image- new figure
Write a rule for the translation ofABCto A′B′C′.Then verify that the transformation is an isometry.
a. • Name the vector and write its component form.
The vertices of ∆LMNare L(2, 2), M(5, 3), and N(9, 1). Translate ∆LMNusing the vector –2, 6.
A boat heads out from point A on one island toward point D on another. The boat encounters a storm at B, 12 miles east and 4 miles north of its starting point. The storm pushes the boat off course to point C, as shown. Write the component form of AB, BC, and CD.
9.2 Using Properties of Matrices • Matrix- a rectangular arrangement of numbers in rows and columns • Element- each number in the matrix • Dimensions- row x column
9.3 Performing Reflections • A reflection in a line (m) maps every point (P) in the plane to a point (P`) so that for each point, one of the following is true:
Rules for Reflections • If (a,b) is reflected in the x-axis, its image is (a,-b). • If (a,b) is reflected in the y-axis, its image is (-a,b). • If (a,b) is reflected in the line y = x, its image is (b,a). • If (a,b) is reflected in the line y = -x, its image is (-b,-a).
You and a friend are meeting on the beach shoreline. Where should you meet to minimize the distance you must both walk?
Find the reflection of PQR in the x- axis using in matrix multiplication. • P(-3,6) Q(-5,3) R(-1,2)
9.4 Performing Rotations • A rotation is an isometry • Center of rotation- a fixed point in which a figure is turned about • Angle of Rotation- the angle formed from rays drawn from the center of rotation to a point and its image
Rules for Rotations • These rules apply for counterclockwise rotations about the origin • a 90o rotation (a,b) (-b,a) • a 180orotation (a,b) (-a,-b) • a 270orotation (a,b) (b,-a)
9.5 Applying Compositions of Transformations • Composition of Transformation- 2 or more transformations are combined to form a single transformation • The composition of 2 (or more) isometries is an isometry.