7 1 Rigid Motion in a Plane Geometry

7. 1 Rigid Motion in a Plane Geometry, NCSCOS: 3. 01 Describe the transformation (translation, reflection, rotation, dilation) of polygons in the coordinate plane in simple algebraic terms.

Objective: � � � Students will understand geometric concepts and applications. Describe the effect of rigid motions on figures in the coordinate plane and space that include rotations, translations, and reflections Identify the three basic rigid transformations.

Essential Question: � How do we identify the three basic rigid transformations in a coordinate plane?

Identifying Transformations �Figures in a plane can be ◦ Reflected ◦ Rotated ◦ Translated to produce new figures. The new figure 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.

Which are basic transformations? � Three basic transformations: 1. 2. 3. 4. � 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.

Examples: Rotation about a point Reflection in a line Translation

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. ”

http: //members. cox. net/fathauerart/Tessellation. Art. html http: //www. tessellations. org/eschergallery 1 thumbs. htm

Example 1: Naming transformations a. b. c. 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 a. 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 b. 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 c. 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

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 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 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 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.

Mappings 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 ◦ C D, B F. E, and

Ex. 3: Preserving Length and Angle Measures In the diagram ∆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. 35°

Ex. 3: Preserving Length and Angle Measures SOLUTION: 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° Sli de 20

http: //hcss. sharepointsite. net/hchs/crawfordb/Geom etry%20 Lessons/Online%20 Text/lgp 07 bbd. pdf Sli de 21

More Illustrations As you look at the image determine the type of translation. Art work Floor tiling N O I T TA RO A R T http: //library. thinkquest. org/16661 /escher/tessellations. 11. html A L NS N O TI http: //www. camaraslate. com/ images/flooring/E 42601. jpg

More images Art work Stained Glass Window N O TI A L S N A R T http: //library. thinkquest. org/16661 /escher/tessellations. 11. html N O TI C E FL RE http: //www. easyboo. com/im ages/tb/IMG_3814 a. JPG

More images Art work C E FL N O TI RE http: //library. thinkquest. org/ 16661/escher/tessellations. 1 1. html Quilting N O TI A T O R www. lisettescountryfabriccreations. com

Transformations on your computer You can perform each of the transformations with the draw tool on your computer. One place to do this is in Word.

An example from Word Translation Rotation (copy and paste) (green circle and drag) Reflection (draw menu and flip horizontal)