2-46. WARM-UP STRETCH Today you will investigate a transformation called a dilation that enlarges or reduces a figure while maintaining its shape. After creating new enlarged polygons, you and your team will explore the relationships that exist among polygons that look alike but are different sizes. Before computers and copy machines existed, it sometimes took hours to enlarge documents or to shrink text to place on small objects like jewelry. A pantograph device (like the one shown below) was often used to duplicate and enlarge written documents and artistic drawings. During this activity, discuss the following questions with your team: ◦ What will the image look like? ◦ What does the image have in common with the original? ◦ What is different about the image? (Be specific.) ◦ How is the image oriented? You will now employ the same geometric principles as a pantograph by using rubber bands to draw enlarged copies of a design.

2.2.1 What do these shapes have in common? September 28, 2015

Objectives CO: SWBAT dilate images. LO: SWBAT investigate the characteristics that the image shares with the original.

This is a type of transformation, called a “dilation”, and the figure drawn using the rubber band chain is an image of the original figure. When a figure is dilated, it is stretched proportionally from a stretch point. The result is an enlarged or reduced figure that looks the same as the original figure. The stretch point is called the “point of dilation” or the “center of dilation”. What is the relationship between the image and the original? Create “What if… ?” questions that can spur further investigation. Such as… ◦ What happens if the point of dilation is closer to the original figure? What if it is moved farther away? ◦ What if the point of dilation is inside of the figure? ◦ What would change if we used four rubber bands? Or five? ◦ How can we change the orientation of the image?

2-47. Stretching a figure as you did in problem 2-46 is a transformation called a dilation. What does a dilated image have in common with the original figure? To answer this question, your team will create dilations that you can measure and compare. Locate the polygon shown in Diagram #1. a. Imagine that a rubber band chain is stretched from the origin so that the first knot traces the perimeter of the original polygon. Dilate the polygon from the origin by imagining a chain of 2(#1), 3(#2), 4(#3), or 5(#4) rubber bands to form A'B'C'D'. b. Carefully trace your dilated polygon from Diagram #1 on tracing paper and compare it to your teammates’ polygons. How are the four dilation images different? How are they the same? As you investigate, make sure you compare both the angle measures and the side lengths of the polygons. When an image is dilated, the angles are congruent and the sides are proportional (100 dilations = 100 times the side length)

2-47. Stretching a figure as you did in problem 2-46 is a transformation called a dilation. What does a dilated image have in common with the original figure? To answer this question, your team will create dilations that you can measure and compare. Locate the polygon shown in Diagram #1. c. Locate Diagram #2 on the resource page. Dilate it by a factor of 3 (with three rubber bands) using point D as the center of dilation. Do your observations from part (b) still apply? What conjectures can you make about dilating any polygon? Be prepared to share your ideas with the class.

2-48. SIMILAR POLYGONS Is one of these sailboats different from the others? Are all four sailboats unique? Do any of the sailboats have something in common? We use the word “similar” to describe two polygons that look alike but may be different sizes. More specifically, similar polygons can be mapped onto each other with a sequence of rigid transformations and a dilation.

2.2.1 What do these shapes have in common? September 29, 2015

Objectives CO: SWBAT use dilations in similar polygons. LO: SWBAT investigate the characteristics that the image shares with the original.

Your Task: For each set of polygons below, three are similar (that is, they are related through a sequence of transformations including dilation), and one is an exception. What is the exception in each set of polygons? Use tracing paper to help you answer each of these questions for both sets of shapes below: ◦ Which polygon appears to be the exception? What makes that polygon different from the others? ◦ What do the other three polygons have in common? ◦ What sequence of transformations would map one polygon onto another?

2-49. Examine the triangles. They are drawn to scale. a. Are they similar? Justify your answer. Use tracing paper to help. b. Which of the following statements are correctly written and which are not? Note that more than one statement may be correct. Discuss your answers with your team. i. Δ DOG ~ Δ CAT ii. Δ DOG ~ Δ CTA iii. Δ OGD ~ Δ ATC iv. Δ DGO ~ Δ CAT

2-50. LEARNING LOG - Dilations & Similar Polygons Write an entry in your Learning Log about what you learned about dilation and similar polygons. Copy the polygon diagram into your Learning Log. If the larger polygon is a dilation of the smaller polygon, what relationships do the corresponding angles and sides have? What characteristics of the larger polygon would you verify to determine that it is indeed a dilation of the smaller polygon? Where is the point of dilation?

2-57. Find your work from problem The shapes that you created should resemble those on the graph at right a. In problem 2-51, you dilated Δ ABC by a factor of 2 (imagining two rubber bands) to create Δ A'B'C'. Which side of Δ A'B'C' corresponds to CB? Which side corresponds to AB ?2-51 b. What is the relationship of the corresponding sides? Write down all of your observations. How could you determine the side lengths of Δ A'B'C' from the side lengths of Δ ABC? c. Why does A'B' lie directly on AB and A'C' lie directly on AC, but B'C' does not lie directly on BC ? d. Could you determine the side lengths of Δ A'B'C' by adding the same amount to each side of Δ ABC? Try this and explain what happened. e. Monica dilated Δ ABC to get a different triangle. She knows that A"B" is 20 units long. How many times longer than AB is A"B" ? How long is B"C" ? Show how you know.