CCSSM Stage 3 Companion Text Lesson 3-O CCSSM Stage 3 Companion Text Transformations and Congruence
Warm-Up 1. Describe the translation that moves A(–3, 4) to A'(1, 3). 2. Describe the type of reflection that moves B(5, 1) to B'(–5, 1). 3. Describe the type of transformation that occurs when you multiply all x- and y-coordinates by 4. Shifted 4 units right and 1 unit down. Reflected over the y-axis. Dilation with scale factor of 4.
Transformations & Congruence Lesson 3-O Transformations & Congruence Target: Understand the relationship between the pre-image and the image of a transformation.
Vocabulary Congruent Similar Two figures are congruent if they are the exact same shape and the exact same size. DOG CAT Similar Two figures are similar if they are the same shape but different sizes. HOG is similar to PEN C T A D G O DOG CAT is read “triangle D-O-G is congruent to triangle C-A-T.” H G O P E N
Explore! Congruent or Similar? In this Explore! you will draw the image of a triangle given a transformation and then determine if the image is congruent or similar to the original triangle. Step 1 List the four types of transformations on four separate lines of your paper. Step 2 Each type of transformation forms an image that is either similar or congruent to the pre-image. Make a conjecture for each transformation you listed in Step 1 about whether the image will be congruent or similar to the pre-image. Step 3 Use ABC for Steps 4 through 7: A(2, 5), B(4, 3), and C(3, 0). Use graph paper to make five coordinate planes (each from −10 to 10 on both axes). Graph the pre-image on the first coordinate plane.
Explore! Congruent or Similar? Step 4 Translation. On the second plane, graph AꞌBꞌCꞌ by shifting ABC 3 units left and 2 units up. Step 5 Reflection. On the third plane, graph AꞌBꞌCꞌ by reflecting ABC over the x-axis. Step 6 Dilation. On the fourth plane, graph AꞌBꞌCꞌ using a scale factor of 2. Step 7 Rotation. On the last plane, graph AꞌBꞌCꞌ by rotating ABC 180o clockwise about the origin. For each transformation, what do you notice about the image compared to the pre-image of ABC? Use the word “similar” or “congruent” in your observations. Step 8 Compare your observations with your original conjectures. Did your work in Steps 4-7 support your conjectures? If not, rewrite your conjecture(s) to match your work. ?
Congruence & Similarity with Transformations Reflection The pre-image and its image are congruent . Translation The pre-image and its image are congruent . Rotation The pre-image and its image are congruent . Dilation The pre-image and its image are similar.
Example 1 State the type of transformation described using each transformation rule. Then state if the image will be congruent or similar to its pre-image. a. x – 3 means shifting left 3 units and y + 4 means shifting up 4 units. This is a TRANSLATION. Translations form CONGRUENT figures.
Example 1 (Cont.) Describe the type of transformation described using each transformation rule. Then state if the image will be congruent or similar to its pre-image. b. A transformation that multiplies the coordinates of a figure by a scale factor (in this case, 4) is a DILATION. Dilations form SIMILAR figures.
Example 2 EFG was formed by a single transformation of MNP. a. Are EFG and MNP congruent or similar? b. Write a translation rule that maps MNP onto EFG. a. EFG and MNP are congruent because they are the same shape and same size. b. MNP is moved 2 units to the right and 6 units down to make EFG.
Transformation Rules Reflection over the y-axis The x-coordinate changes sign. Reflection over the x-axis The y-coordinate changes sign. Dilation Shown by multiplying the x- and y-coordinates by the scale factor. Examples: scale factor of 5 scale factor of
Exit Problems Describe the transformation given by each rule. State whether the transformation creates similar or congruent figures. 1. (x, y) (x, y + 8) 2. (x, y) (x, y) 3. (x, y) (x + 2, y – 1) 4. (x, y)
Communication Prompt How can you use transformations to show that two figures are similar?