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Transformations andRigid oficasEssential question: How do you identify transformations that are rigid motions?

Introducing Transformations

A ~f~~f9~ii~l(),;iis a function that changes the position, shape, and! or size of a figure.The inputs for the function are points in the plane; the outputs are other points in theplane. A figure that is used as the input of a transformation is the P'~~~~~g~The outputis the ii.liag~.

For example, the transformation T moves point A to point A', Point Ais the pre-image, and A' is the image.You can use function notationto write T(A) = A'. Note that a transformation is sometimes called amapping. Transformation T maps point A to point A',

Coordinate notation is one way to write a rule for a transformationon a coordinate plane. The notation uses an arrow to show how thetransformation changes the coordinates of a general point, (x, y).

For example, the notation (x,y) ~ ex + 2, y - 3) means that the transformationadds 2 to the x-coordinate of a point and subtracts 3 from its y-coordinate. Thus, thistransformation maps the point (6, 5) to the point (8,2).

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ta. Explain how to identify the pre-image and image in T(E) = F.

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!! 1b" Consider the transformation given by the rule (x, y) -+ ex + I, y + 1). What is thel domain of this function? "Vhat is the range? Describe the transformation.

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Transformation T maps points in the coordinate plane by moving themvertically up or down onto the x-axis. (Points on the x-axis are unchanged bythe transforrnation.) Explain how to use coordinate notation to write a rule fortransformation 1'.

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Unit 2 35

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Lesson 1

Investigate the effects of various transformations on the given right triangle.

@ Use coordinate notation to help you find the image of each vertex of the triangle .

•• Plot the images of the vertices.

" Connect the images of the vertices to draw the image of the triangle.

'A (x, y) ~ ex - 4, Y + 3)

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2;'L A transformation preserves distance if the distance between any two points of thepre-image equals the distance between the corresponding points of the image.Which of the above transformations preserve distance'?

2b. A transformation preserves angle measure if the measure of any angle of thepre-image equals the measure of the corresponding angle of the image.Which of the above transformations preserve angle measure?

Unit 2 36 Lesson 1

A iim:Wm:9j!!!~ (or isometry) is a transformation that changes the position of a figurewithoutchanging the size or shape of the figure.

The figures show the pre-image (6ABC) and image{L::!.A'B'C)under a transformation.Determine whether the transformation appears to be a rigid motion. Explain.

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The transformation does not change the size or shape of the figure

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The transformation changes the shape of the figure.

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How could you use tracing paper or a transparency to help you identifyrigid motions?

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I 3b. Which of the transformations on the previous page appear to be rigid motions?

<..Unit 2 37 Lesson 1

Rigid motions have some important properties. These are summarized below .

•• Rigid motions preserve distance." Rigid motions preserve angle measure.

,1 "Rigid motions preserve betweenness,1 {I Rigid motions preserve collinearity.

The above properties ensure that if a figure is determined by certain points, then its imageafter a rigid motion is also determined by those points. For example, 6.4.BC is determinedby its vertices, points A, B, arid C. The image of 6ABCafter a rigid motion is the triangledetermined by A', B', and C.

Draw the image of the triangle under the given transformation. Then tellwhether the transformation appears to be a rigid motion.

1. (x, y) --' (x + 3, y)

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Unit 2

2. (x, y) -+ (3x, 3y)

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38

3. (x, y) -+ (x, -y)

6. ex, y) -+ (x - 4, Y - 4)

Lesson 1