6-4 Dilations. SOLUTION: The dilation is. Multiply the coordinates of each vertex by 2. The coordinates after the dilation are C (2, 8), A plane.

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1 Find the coordinates of the vertices of each figure after a dilation with the given scale factor k. Then graph the original image and the dilation. 1. C(1, 4), A(2, 2), T(5, 5); k = 2 The dilation is. Multiply the coordinates of each vertex by 2. The coordinates after the dilation are C (2, 8), A plane., and T (10, 10). Graph both triangles on the same coordinate esolutions Manual - Powered by Cognero Page 1

2 2. R(1, 1), S(1, 7), T(5, 7), U(5, 1); k = The dilation is. Multiply the coordinates of each vertex by. The coordinates after the dilation are. Graph both quadrilaterals on the same coordinate plane. esolutions Manual - Powered by Cognero Page 2

3 3. A graphic designer created a logo on -by 11-inch paper. In order to be placed on a business card, the logo needs to be inches by inches. What is the scale factor of the dilation? Write a ratio comparing the side lengths of the logo and the business card. The scale factor of the dilation is. 4. Darian wants to build a regulation-size pool table that is 9 feet in length. The plans he ordered are 18 by 36 inches. What is the scale factor of the dilation he must use to build the regulation pool table? Write a ratio comparing the side lengths of the plans and the actual table. First convert 36 inches to feet = 3 feet The scale factor of the dilation is 3. esolutions Manual - Powered by Cognero Page 3

4 5. A triangle has vertices A( 2, 3), B(0, 0), and C(1, 1). a. Find the coordinates of the triangle if it is reflected over the x-axis, then dilated by a scale factor of 3. b. Find the coordinates if the original triangle is dilated by a scale factor of 3, then reflected over the x-axis. c. Are the two transformations commutative? Explain. a. Find the coordinates after the reflection over the x-axis by multiplying the y-coordinates of each vertex by 1. The coordinate for the triangle after a reflection over the x-axis are A ( 2, 3), B (0, 0), and C (1, 1). To dilate ΔA B C by a scale factor of 3, multiply the coordinates of each vertex by 3. The coordinates of ΔA B C are A ( 6, 9), B, and C (3, 3). b. To dilate ΔABC by a scale factor of 3, multiply the coordinates of each vertex by 3. The coordinates of ΔA B C after the dilation are A, B (0, 0), and C (3, 3). To find the coordinates after the reflection over the x-axis by multiplying the y-coordinates of each vertex by 1. The coordinates of ΔA B C are A ( 6, 9), B, and C (3, 3). c. Yes; Sample answer: since the coordinates of the answers to Exercises a and b are the same, the order in which you perform them does not matter. esolutions Manual - Powered by Cognero Page 4

5 6. Model with Mathematics In each part of the graphic organizer, sketch an image of pentagon MNOPQ after a dilation within the given parameters. A dilation with a scale factor of k will be a reduction when 0 < k < 1. So, draw pentagon M'N'O'P'Q' in the lefthand section the same shape as pentagon MNOPQ but smaller. A dilation with a scale factor of k will be the same as the original figure when k = 1. Draw pentagon M'N'O'P'Q' in the middle section the same shape and size as pentagon MNOPQ. A dilation with a scale factor of k will be an enlargement when k > 1. So, draw pentagon M'N'O'P'Q' in the right-hand section the same shape as pentagon MNOPQ but larger. esolutions Manual - Powered by Cognero Page 5

6 7. Make a Conjecture A figure has a vertex at the point ( 4, 6). The figure is dilated with the center at the origin with a scale factor of 5. The resulting image is then dilated with a scale factor of. a. What are the coordinates of the vertex in the final image? b. How do they compare with those of the original image? c. Can you predict the scale factor of a compound dilation? Explain. a. First find the coordinates of the vertex after a dilation of scale factor 5. Next, find the coordinates of the vertex after a dilation of scale factor. The coordinates of the vertex after both dilations is ( 12, 18). b. The final coordinates are three times the original coordinates. c. Sample answer: Yes; multiply the scale factors of each dilation to find the scale factor of the final dilation. 8. Persevere with Problems The coordinates of two triangles are shown in the table. Is ΔWXY a dilation of ΔABC? Explain. ΔWXY is not a dilation of ΔABC. Sample answer: both coordinates of all the points must be multiplied by the same scale factor. The x-coordinates are multiplied by 4, but the y-coordinates are only multiplied by Persevere with Problems The algebraic representation of a dilation is. If the dilation is an enlargement, give three possible values of a. Sample answer:,,. If a is replace with. then So, this is an enlargement with a scale factor of 3. In this dilation any value of a between 0 and 1 will create a enlargement. esolutions Manual - Powered by Cognero Page 6

7 Find the coordinates of the vertices of each figure after a dilation with the given scale factor k. Then graph the original image and the dilation. 10. R(5, 5), S(5, 10), T(10, 10), U(10, 5); k = The dilation is. Multiply the coordinates of each vertex by. The coordinates after the dilation are R (2, 2), S same coordinate plane., T (4, 4), and U (4, 2). Graph both quadrilaterals on the esolutions Manual - Powered by Cognero Page 7

8 11. V( 3, 4), X( 2, 0), W(1, 2); k = 3 The dilation is. Multiply the coordinates of each vertex by 3. The coordinates after the dilation are V ( 9, 12), X coordinate plane., and W (3, 6). Graph both triangles on the same 12. To place a picture in his class newsletter, Joaquin must reduce the picture by a scale factor of 0.3. Find the dimensions of the reduced picture if the original is 15 centimeters wide and 10 centimeters high. Multiply the dimensions of the original picture by the scale factor cm 0.3 = 4.5 centimeters 10 cm 0.3 = 3 centimeters The dimensions of the reduced picture are 4.5 centimeters by 3 centimeters. esolutions Manual - Powered by Cognero Page 8

9 13. Multiple Representations Triangle XYZ has vertices X(0, 0), Y(3, 1) and Z(2, 3). a. Numbers Find the coordinates of the image of ΔXYZ after a dilation with a scale factor of 2. b. Algebra Graph ΔXYZ and the image on the coordinate plane. c. Words Describe the locations of ΔXYZ and ΔX'Y'Z' using transformations. a. The dilation is (x, y) ( 2x, 2y). Multiply the coordinates of each vertex by 2. X(0, 0) ( 2 0, 2 0) (0, 0) Y(3, 1) ( 2 3, 2 1) ( 6, 2) Z(2, 3) ( 2 2, 2 3) ( 4, 6) The coordinates after the dilation are X'(0, 0), Y'( 6, 2), and Z'( 4, 6). b. Graph ΔXYZ and ΔX'Y'Z' on the same coordinate plane. c. ΔX'Y'Z' is the image of ΔXYZ after a dilation of 2 and a rotation of 180 about the origin. esolutions Manual - Powered by Cognero Page 9

10 14. Triangle RST is dilated so that the image of point T is T'(6, 6). Draw triangle R'S'T'. What is the scale factor of the dilation? Does the dilation represent an enlargement or a reduction? Explain how you found your answer. Select one of the points and find the scale factor. Point T is located at (4, 4) and point T is located at (6, 6). Since ΔR S T is larger than ΔRST, it is an enlargement and the scale factor will be greater than 1. Write a ratio comparing the x-coordinates of the two vertices. The scale factor of the dilation is or 1.5. Sample answer: Compare point T(4, 4) and point T (6, 6). To transform point T(4, 4) to T (6, 6), you multiply by each coordinate by 1.5. So the scale factor is 1.5. Since 1.5 > 1, the dilation represents an enlargement. esolutions Manual - Powered by Cognero Page 10

11 15. Squares A and B are related by a dilation. Determine if each statement is true or false. a. The scale factor from figure A to figure B is. b. The scale factor from figure B to figure A is. c. The dilation from figure A to figure B is an enlargement. Since square B is smaller than square A, the dilation is a reduction and the scale factor will be between 0 and 1. Write a ratio of the side length of square B to the side length of square A. The scale factor is. a. True b. True c. False 16. A model airplane is built with a wing span of 18 inches. The actual wing span of the airplane is 90 feet. Find the scale. Write a ratio that compare the measurement of the model wing span to the measurement of the actual wing span. Convert feet to inches to simplify the ratio. Then convert inches to feet to write the scale. So, the scale is 1 inch = 5 feet. esolutions Manual - Powered by Cognero Page 11

12 Find the scale factor for each scale in. = 10 ft Write the scale as a ratio. Convert feet to inches. Then simplify the ratio and divide out the common units. So, the scale factor is cm = 2.5 mm Write the scale as a ratio. Convert centimeters to millimeters. Then simplify the ratio and divide out the common units. So, the scale factor is ft = 15 yd Write the scale as a ratio. Convert yards to feet. Then simplify the ratio and divide out the common units. So, the scale factor is. esolutions Manual - Powered by Cognero Page 12

13 20. On a map of Kansas, the scale is 1 in. = 50 miles. Using the scale, complete the table showing the distance between cities. To find the actual distance from Wichita to Topeka, write and solve a proportion using the scale. Let x represent the actual distance. The distance between Wichita and Topeka is miles. To find the map distance from Salina to Kansas City, write and solve a proportion using the scale. Let x represent the map distance. The distance between Salina and Kansas City on the map is 3 inches. esolutions Manual - Powered by Cognero Page 13