7.1 RIGID MOTION IN A PLANE Unit 1B Day 1

DO NOW In the pairs of figures below, the blue figure was transformed to produce the congruent red figure. For each pair, name the corresponding sides. a) GH & JL, HF &JK, FG & KJ b) AB & JK, BC & KL, CD & LM, DE & MN, EA & NJ

IDENTIFYING TRANSFORMATIONS Figures in a plane can be reflected, rotated, or 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. We usually use the prime symbol to denote a transformation. For example, if the preimage is called A, its image would be called A'.

BASIC TRANSFORMATION flip turn slide

EX. 1: NAMING TRANSFORMATIONS 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? a) The transformation is a reflection in the y-axis. You can imagine that the image was obtained by flipping ∆ABC over the y-axis. b) The coordinates of the vertices of the image, ΔA’B’C’, are A’(4, 1), B’(3, 5), and C’(1, 1). c) Yes, ΔABC is congruent to its image ΔA’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.

MAPPINGS You can describe the transformation in the diagram by writing “∆ABC is mapped onto ∆DEF.” Notation: ∆ABC  ∆DEF Order matters: If ΔABC is mapped onto ΔDEF, it implies that A  ____, B  ____, and C  ___.

ISOMETRY An isometry is a transformation that 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. This transformation is not an isometry. The image is not congruent to the preimage. This transformation appears to be an isometry. The blue parallelogram is rotated about a point to produce a congruent red parallelogram.

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 XY and the measure of Z. 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. 4: IDENTIFYING TRANSFORMATIONS You are assembling pieces of wood to complete a railing for your porch. The finished railing should resemble the one below. How are pieces 1 and 2 related? pieces 3 and 4? Pieces 1 and 2 are related by a rotation. Pieces 3 and 4 are related by a reflection.

Ex. 5 Use the graph of the transformation above. ABCDE is the preimage. Figure ABCDE → Figure______. Name and describe the transformation. Name the image of CD. Name the preimage of FJ. Name the coordinates of the preimage of point I. Show that DE and IJ have the same length, using the Distance Formula. FGHIJ Reflection in the y-axis HI AE (2, 3) Both have length √8

Ex. 6 Use the diagrams to complete the statement. ∆ABC→∆ ______ ∆DEF→ ∆______ ∆ ____ → ∆ACB ∆_____→∆ CBA ∆RQP→ ∆______ ∆ ____ → ∆EFD JKL PQR KLJ LKJ FED QRP

CLOSURE Is the transformation below an isometry? Why or why not? No because length is not preserved.