6.2 EXERCISES HOMEWORK = See WORKED-OUT SOLUTIONS KEY Exs. 5, 15, 19, and 27 X = STANDARDIZED TEST PRACTICE Exs. 2, 6, 10, 17, 26, and 30 SKILL PRACTICE 1. VOCABULARY Draw an example of a dilation. 2. *WRITING Describe the results of a similarity transformation. EXAMPLE 1 DESCRIBING DILATIONS Describe the dilation that moves the blue figure : for Exs. 1-6 onto the red figure. 3. 5. 10 6. * WRITING Jenny described a transformation in the coordinate plane using the notation (x, y) -> (3x, 3y). Explain why her transformation is a dilation with center at the origin. EXAMPLE 2 DRAWING SIMILARITY TRANSFORMATIONS Copy the figure. Draw an example : for Exs. 7-9 of the given similarity transformation of the figure with center O. 7. 0 8. 9. 0 dilation dilation then reflection dilation then rotation EXAMPLE 3 10. * MULTIPLE CHOICE Which of the following transformations does not : for Ex. 10-12 involve dilation? A B C D 370 Chapter 6 SimilarityIDENTIFYING DILATIONS The red figure is the image of the blue figure under a transformation. Tell whether the transformation involves a dilation. If so, give the scale factor of the dilation. 11. 12. 30 40 COMBINING TRANSFORMATIONS Coordinates of the vertices of a preimage and image figure are given. Describe the transformations that move the first figure onto the second. 13. O(0, 0), B(0, 3), C(2, 0); O(0, 0), D(0, -6), E(4, 0) 14. O(0, 0), P(0, 8), Q(6, 0); O(0, 0), R(4, 0), S(0, -3) 15. J(-4, 0), K(0, 4), L(6, 0); L(6, 0), M(0, 6), N(-9, 0) 16. A(-6, 0), B(0, 6), C(9, 0), D(0, -15); W(0, -2), X(-2, 0), Y(0, 3), Z(5, 0) 17. SHORT RESPONSE A fractal called the Sierpinski triangle can be imagined by thinking of an infinite sequence of stages, the first of which are shown below. Calculate the side lengths of the light blue triangles in Stages 1, 2, and 3. Then describe the 3 dilations you can apply to any stage to generate the next stage. 16 16 16 stuolson Stage 0 Stage 1 Stage 2 Stage 3 SIMILARITY OF CIRCLES Prove the circles are similar by finding a center and scale factor of a dilation that moves the blue circle onto the red circle. 18. concentric 19.) intersecting 1:23 3.5. 20. tangent Becam many 21. not intersecting 12 6.2 Relate Transformations and Similarity 37122. CHALLENGE The dilation of a rectangle has a scale factor of 4 to 1. a. Describe the effect of the dilation on the perimeter of the rectangle. b. Describe the effect of the dilation on the area of the rectangle. PROBLEM SOLVING EXAMPLE 4 23. TEXTILES A cloth purse maker uses a pattern for a small bag. The maker : for Exs. 23-28 wants to start making a similar bag that is exactly twice as big as the small one and has the same design. Explain how you might efficiently create the design for the large bag. 24. GRAPHIC ARTS Describe how using an overhead projector involves dilations when creating a larger image that can be traced onto a poster. 25. CRAFTS A rug design uses similar triangles that become larger on one side while getting smaller on the other side. What transformations are used to move figure A onto figure B? 26. * OPEN-ENDED Create a design using a combination of dilations and other transformations. 27. TOY IDEA A flashlight has a removable cap with a picture on it. When the flashlight is on, the bug can be projected onto a wall. Using the information in the diagram below, find the scale factor of the dilation. Then calculate the height of the bug projected onto the wall. 2 cm 2.5 cm 10 cm 28. ANIMATION You are designing an animated lightning bolt on a computer screen. You want the distance of each bolt from P to be = greater than the distance of the previous bolt. Describe the transformation that moves each bolt to the next larger bolt. 29. CHALLENGE Describe a method for finding the center and scale factor of a dilation that moves a given circle onto another given circle with a different radius. 372 Chapter 6 Similarity30. EXTENDED RESPONSE The figure below shows an architect's initial layout of the floor plan for a house. The scale is ,, inch = 1 foot. a. Interpret How is a dilation of the floor plan used in the house construction? 7.5 in. 2.5 in. 4 in 6 in What is the scale factor of the dilation? 5 in. BEDROOM DEN BEDROOM | 5 in. STORAGE b. Model Explain how to find the actual 2 size of the living room in the house BATH using the floor plan. 3 in. c. Calculate Find the actual size of each 4 in. 5 in. LIVING KITCHEN 4 in. room in the house. 10 in. 8 in. 31. OPEN-ENDED A transformation in the coordinate plane is described using the notation (x, y) - (2x, 5y). Explain why the transformation is not a similarity transformation by showing how it affects the angles and sides of a polygon. ty (ka, kb) 32. CHALLENGE A dilation of a triangle is shown, in which the center of dilation lies in the interior of the triangle. (ke, kf) ( a, b) ( e, f) a. Use the slope formula to show that corresponding sides are parallel. b. The black rays in the diagram are transversals that Nic, d intersect parallel segments. Prove the angles marked are congruent. Explain why the angles of the preimage ( kc , kd ) and image triangles are preserved under the dilation. See EXTRA PRACTICE in Student Resources 373