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7.1 Rigid Motion in a Plane. Geometry Mrs. Spitz Spring 2005. Objectives. Identify the three basic rigid transformations. Use transformations in real-life situations such as building a kayak in example 5. Assignment: Pp. 399-400 #1-39. Identifying Transformations.

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7 1 rigid motion in a plane l.jpg

7.1 Rigid Motion in a Plane

Geometry

Mrs. Spitz

Spring 2005


Objectives l.jpg
Objectives

  • Identify the three basic rigid transformations.

  • Use transformations in real-life situations such as building a kayak in example 5.

    Assignment:

  • Pp. 399-400 #1-39


Identifying transformations l.jpg
Identifying Transformations

  • Figures in a plane can be

    • Reflected

    • Rotated

    • Translated

  • To produce new figures. The new figures is called the IMAGE. The original figures is called the PREIMAGE. The operation that MAPS, or moves the preimage onto the image is called a transformation.


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What will you learn?

  • Three basic transformations:

    • Reflections

    • Rotations

    • Translations

    • And combinations of the three.

  • For each of the three transformations on the next slide, the blue figure is the preimage and the red figure is the image. We will use this color convention throughout the rest of the book.


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Copy this down

Rotation about a point

Reflection in a line

Translation


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Some facts

  • Some transformations involve labels. When you name an image, take the corresponding point of the preimage and add a prime symbol. For instance, if the preimage is A, then the image is A’, read as “A prime.”


Example 1 naming transformations l.jpg

Use the graph of the transformation at the right.

Name and describe the transformation.

Name the coordinates of the vertices of the image.

Is ∆ABC congruent to its image?

Example 1: Naming transformations


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Name and describe the transformation.

The transformation is a reflection in the y-axis. You can imagine that the image was obtained by flipping ∆ABC over the y-axis/

Example 1: Naming transformations


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Name the coordinates of the vertices of the image.

The cordinates of the vertices of the image, ∆A’B’C’, are A’(4,1), B’(3,5), and C’(1,1).

Example 1: Naming transformations


Example 1 naming transformations10 l.jpg

Is ∆ABC congruent to its image?

Yes ∆ABC is congruent to its image ∆A’B’C’. One way to show this would be to use the DISTANCE FORMULA to find the lengths of the sides of both triangles. Then use the SSS Congruence Postulate

Example 1: Naming transformations


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ISOMETRY

  • An ISOMETRY is a transformation the preserves lengths. Isometries also preserve angle measures, parallel lines, and distances between points. Transformations that are isometries are called RIGID TRANSFORMATIONS.


Ex 2 identifying isometries l.jpg

Which of the following appear to be isometries?

This transformation appears to be an isometry. The blue parallelogram is reflected in a line to produce a congruent red parallelogram.

Ex. 2: Identifying Isometries


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Which of the following appear to be isometries?

This transformation is not an ISOMETRY because the image is not congruent to the preimage

Ex. 2: Identifying Isometries


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Which of the following appear to be isometries?

This transformation appears to be an isometry. The blue parallelogram is rotated about a point to produce a congruent red parallelogram.

Ex. 2: Identifying Isometries


Mappings l.jpg

You can describe the transformation in the diagram by writing “∆ABC is mapped onto ∆DEF.” You can also use arrow notation as follows:

∆ABC  ∆DEF

The order in which the vertices are listed specifies the correspondence. Either of the descriptions implies that

A  D, B  E, and C  F.

Mappings


Ex 3 preserving length and angle measures l.jpg

In the diagram writing “∆PQR is mapped onto ∆XYZ. The mapping is a rotation. Given that ∆PQR  ∆XYZ is an isometry, find the length of XY and the measure of Z.

Ex. 3: Preserving Length and Angle Measures

35°


Ex 3 preserving length and angle measures17 l.jpg

SOLUTION: writing “

The statement “∆PQR is mapped onto ∆XYZ” implies that P  X, Q  Y, and R  Z. Because the transformation is an isometry, the two triangles are congruent.

So, XY = PQ = 3 and mZ = mR = 35°.

Ex. 3: Preserving Length and Angle Measures

35°


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