, ; Obtain a Lesson Resource Page from your teacher. On it, find the quadrilateral shown in Diagram # 1 at right. Diagram #1

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1 3-2. Stretching a figure as ou did in problem 3-1 is another transformation called a dilation. When a figure is dilated from a point, the result is a similar figure. How are dilated figures related to their original figures? That is, what do similar figures have in common? To answer these questions and to develop a definition of "similar," our team will need to create dilations that ou can measure and compare., ;,._;--- ; x a. Obtain a Lesson Resource Page from our teacher. On it, find the quadrilateral shown in Diagram # 1 at right. Diagram #1! Dilate (stretch) the quadrilateral from the origin b a factor of 2,3,4. or 5 to form A' B' C'D'. Each team member should pick a different enlargement factor. You ma want to imagine that our rubber band chain is stretched from the origin so that the knot traces the perimeter of the original figure. For example, if our job is to stretch ABeD b a factor of 3, then A' would be located as shown in Diagram #2 at right. Diagram #2 b. Carefull cut out our enlarged figure and compare it to our teammates' figures. How are the four enlargements different? How are the the same? As ou investigate, make sure ou compare both angles and side lengths of the similar figures. Be read to report our conclusions to the class. Chapter 3: Justification and Similarit 303

2 ~------~-..--.~ ~ ~ WHICH SHAPE IS THE EXCEPTION? Sometimes figures look the same and sometimes the look ver different. What characteristics make figures alike so that ou can sa that the are the same shape? How are figures that look the same but are different sizes related to each other? Understanding these relationships will allow us to know if figures that appear to have the same shape actuall do have the same shape. Your Task: For each set of figures below, three are similar (meaning that the are related through a sequence of transformations including dilation), and one is an exception. Find the exception in each set of figures. Use tracing paper to answer each of these questions for both sets of shapes below: Which figure appears to be the exception? What makes that shape different from the others? What do the other three shapes have in common? Are there commonalities in the angles? Are there differences? Are there commonalities in the sides? Are there differences? a. b LEARNING LOG Write an entr in our Learning Log about the characteristics that figures with different sizes need to have in order to maintain the same shape. Add our own diagrams to illustrate the description. Title this entr "Same Shape, Different Size" and include toda's date. 304 Core Connections Geometr

3 --_._ THODS AND MEANINGS C/) w o Z I «~ Dilations The transformations ou studied in Chapter 1 (translations, rotations, and reflections) are called rigid transformations because the all maintain the size and shape of the original figure. Enlargement However, a dilation is a transformation that maintains the shape of a figure but multiplies its lengths b a chosen factor. In a dilation, a figure is stretched proportionall from a particular point, called the point of dilation or stretch point. For example, in the diagram at right, MBC is dilated to form M' B'C'. Notice that while a dilation changes the size and location of the original figure, it does not rotate or reflect the original. While lengths can change, angles do not change under a dilation. Note that if the point of dilation is located inside a shape, the enlargement encloses the original, as shown at right. Point of Dilation ~~&~====,.. ' P"evie Plot triangle ABC formed with the points A(O,0), B(3, 4), and C(3, 0), on graph paper. Use the method used in problem 3-2 to enlarge it from the origin b a factor of 2 (using two "rubber bands"). Label this new triangle A' B' C'. a. What are the side lengths of the original triangle, MBC? b. What are the side lengths of the enlarged triangle, M' B' C'? c. Find the area and the perimeter of M'B'C'. Chapter 3: Justification and Similarit 305

4 3-6. Solve each equation below for x. Show all work and check our answer b substituting it back into the equation and verifing that it makes the equation true. a..l.=6 3 b. 5x+9 = 12 2 c. L d..i _ 20 x Examine the triangle at right. a. Estimate the measure of each angle of the triangle at right. b. Given onl its shape, what is the best name for this triangle? 3-8. On graph paper, graph line MU if M(-I, 1) and U(4, 5). a. Find the slope of MU and write an equation for the line. b. Find tl1v (the distance from M to U). c. Are there an similarities to the calculations used in parts (a) and (b)? An differences? 3-9. Examine each diagram below. Identif the error in each diagram. a. 3m~ b. ~4cm 5cm~ c. 65 5m 12 em 55 7 in Rewrite the statements below into conditional ("If..., then... ") form. a. All equilateral triangles have rotation smmetr. b. A rectanzle is a parallelogram. _ c. The area of a trapezoid is half the sum of the bases multiolied b the height. 306 Core Connections Geometr

5 Lesson Resource Page Dilation Chapter 3: Justification and Similarit 397