Rising, Running, Stepping, Scaling Dilating Figures on the Coordinate Plane

Transcription

1 Rising, Running, Stepping, Scaling Dilating Figures on the Coordinate Plane WRM UP Scale up or scale down to determine the value of the variable in each equivalent ratio : : z. : 5 5 a : 3 LERNING GOLS Dilate figures on a coordinate plane. Understand the dilation of a figure on the coordinate plane as a scaling up or scaling down of the coordinates of the figure. Describe how a dilation of a figure on a coordinate plane affects the coordinates of a figure. Distinguish between a dilation centered at the origin and a dilation not centered at the origin : 5 :. 9.9 : 5 99 : p You have used transformations called dilations to create similar figures. How can ou use coordinates to determine whether two figures are similar? LESSON : Rising, Running, Stepping, Scaling M1-15

2 Getting Started The Escalator or the Stairs ob is riding an escalator. The escalator starts at (, ) and drops ob off at (1, ). 1. Use the coordinate planes given to represent ob's journe. a. Draw a line to show ob's path on the escalator. Think about equivalent ratios, scaling up, and scaling down. b. lice takes the stairs. Draw steps starting at the origin that will take lice to the same location as ob. Make all of the steps the same.. 1, O O Escalator it3 Stairs. How is taking the stairs similar to riding the escalator? How is it different? Eplain our reasoning. Escalator is Steps are a smooth path repeating up and greggs 3. Compare the steps that ou designed for lice with our classmates steps. How are these steps similar to our steps? M1-1 TOPIC : Similarit

3 CTIVITY.1 Scaling Up and Down on the Coordinate Plane NOTES You know that a translation moves a point along a line. sequence of repeated horizontal and/or vertical translations also moves a point along a line. You can use this fact to dilate figures. WORKED EXMPLE o Dilate C b a scale factor of 3 using the origin as the center of dilation. Let s start b dilating Point, which is located at (, 1). In other words, Point is translated from the origin units right and 1 unit up C To dilate point b a scale factor of 3, translate Point b three repeated sequences: units right and 1 unit up from the origin. 1. Describe the repeated translations ou can use to scale point and point C. Then plot point ' and point C' on the coordinate plane in the worked eample. a. point to point 9 b. point C to point C9 up 3 over 3 over up 1. Draw 99C9 on the coordinate plane in the eample. Is C similar to 99C9? Eplain our reasoning. Yes because the have diff size same shape LESSON : Rising, Running, Stepping, Scaling M1-17

4 WORKED EXMPLE Dilate DEF b a scale factor of 1 using the origin as the center of dilation D D E 1 3 Point D is translated from the origin units right and units up (, ). This is the same as four translations of 1 unit right and 1 unit up. F 1 Therefore, scaling point D to (1, 1) represents a dilation b a scale factor of 1. How do the side lengths and angles of the triangles compare? 3. Determine the coordinates of points E9 and F9. Eplain how ou determined our answers. Then, draw D9E9F9 on the coordinate plane in the eample. E C, F 3,1 D C l. Is DEF similar to D9E9F9? Eplain our reasoning. l Yes the have the same shape but diff size 5. How does dilating a figure, using the origin as the center of dilation, affect the coordinates of the original figure? Make a conjecture using the eamples in this activit. E Eats Ek's M1-1 TOPIC : Similarit

5 CTIVITY. Using the Origin as the Center of Dilation Road signs maintain a constant scale, regardless of whether the are on the road or in the drivers manual. This sign indicates that the road is bending to the left. 1. Dilate the figure on the coordinate plane using the origin (, ) as the center of dilation and a scale factor of 3 to form a new figure D C p FT bigger D C. List the ordered pairs for the original figure and for the new figure. How are the values in the ordered pairs affected b the dilation? i 3. Compare and contrast the corresponding angles and corresponding side lengths of the new figure and the original figure. corresponding angles are Corresponding s o as E side length TreffTnae I 3 LESSON : Rising, Running, Stepping, Scaling M1-19

6 Let s consider a different road sign. This sign indicates that the road proceeds to the right.. Dilate the figure on the coordinate plane using the origin (, ) as the center of dilation and a scale factor of 1 to form a new figure FF Z fx W X Y XI 9 5. List the ordered pairs for the original figure and for the new figure. How are the values in the ordered pairs affected b the dilation? w, w,5 X, 5, Y, Y C,3 Z, 3,7. Compare and contrast the corresponding angles and corresponding side lengths of the original figure and the new figure. corresponding angles are I Eg ae L M1-13 TOPIC : Similarit

7 CTIVITY.3 Using a Point on the Figure as a Center of Dilation NOTES You can use an point as the center of dilation. The center of dilation can be on the figure, inside the figure, or outside the figure. 1. Consider C C n 1T a. Dilate C using point C as the center of dilation and a scale factor of 3 to form 99C9. Eplain how ou determined the coordinates of the dilated figure. Find the distance from C to and melt it b Same for c to b. What are the coordinates of points 9, 9 and C9? C 3 T C 3,15 C 7,3 5,3 C C 3,3 c C 3,3 cannotmd.b LESSON : Rising, Running, Stepping, Scaling M1-131

8 . Consider Quadrilateral CD pr D C c a. Dilate Quadrilateral CD using point C as the center of dilation and a scale factor of 1 to form Quadrilateral 99C9D9. Eplain how ou determined the coordinates of the dilated figure. b. What are the coordinates of points 9, 9, C9, and D9?,13 11,,7 11,7 1,7 1,7 DC,3 c. How are the coordinates of a figure affected b a dilation that is not centered at the origin? cannot simpl P 9,5 divide b because the center of dilation is not at the origin M1-13 TOPIC : Similarit

9 CTIVITY. Using a Point Inside or Outside the Figure as a Center of Dilation In this activit, ou will eplore different center points for dilation to understand how the coordinates of a figure are affected b dilations. 1. Dilate Figure PQRS b a scale factor of 3 using the point (, ) as the center of dilation. Determine the coordinates of Figure P9Q9R9S9 and draw the approimate dilation on the coordinate plane. P Q S R LESSON : Rising, Running, Stepping, Scaling M1-133

10 NOTES. Dilate Figure PQRS b a scale factor of 3 using the point (, ) as the center of dilation. Determine the coordinates of Figure P9Q9R9S9 and draw the approimate dilation on the coordinate plane. P Q S R 3. How are the coordinates of a figure affected b a dilation that is not centered at the origin? How do ou think ou can modif our original conjecture? dilate ll 3 with scale, factor i If the dilation of a figure is centered at the origin, ou can multipl the coordinates of the points of the original figure b the scale factor to determine the coordinates of the new figure. To determine the dilation of a figure not centered at the origin, ou can follow these steps: Subtract the - and -coordinates of the center from the - and -coordinates of each point. Multipl the new coordinates of each point b the scale factor. dd the - and -coordinates of the center to the new - and -coordinates of each point. M1-13 TOPIC : Similarit

11 . Determine the dilation of each triangle using the information given. Verif our answer on the coordinate plane. O a. Center: (3, 3) Scale factor: O c C X b. Center: origin Scale factor: 3 c. Center: (1, 3) Scale factor: C c Tif C center g 1,,, 13, C,3 c'cop LESSON : Rising, Running, Stepping, Scaling M1-135

12 NOTES TLK the TLK Location, Location, Location nswer each question to summarize what ou know about dilating figures on the coordinate plane. Use our answers to plan a presentation for our classmates that demonstrates what ou learned in this lesson. 1. What strategies can ou use to determine if two figures are similar when the are: a. located on a coordinate plane? b. not located on a coordinate plane?. How does the location of the center of dilation affect the coordinates of the dilated figure? 3. Describe how ou can determine whether two figures on the coordinate plane are similar using just their coordinates and the center of dilation. M1-13 TOPIC : Similarit

13 ssignment Write In our own words, eplain how to dilate a figure on the coordinate plane using repeated translations. Use eamples with scale factors less than and greater than 1 to illustrate our eplanation. Remember If the dilation of a figure is centered at the origin, ou can multipl the coordinates of the points of the original figure b the scale factor to determine the coordinates of the new figure. To determine the dilation of a figure not centered at the origin, ou can follow these steps: Subtract the - and -coordinates of the center from the - and -coordinates of each point. Multipl the new coordinates of each point b the scale factor. dd the - and -coordinates of the center to the new - and -coordinates of each point. Practice 1. Graph Triangle XYZ with the coordinates X (, 17), Y (17, 17), and Z (17, ) a. Reduce Triangle XYZ on the coordinate plane using the point Y as the center of dilation and a scale factor of 1 to form Triangle X9YZ9. 3 b. What are the coordinates of points X9 and Z9? LESSON : Rising, Running, Stepping, Scaling M1-137

14 . Dilate Triangle QRS on the coordinate plane using the origin (, ) as the center of dilation and a scale factor of 3 to form Triangle Q9R9S9. Label the coordinates of points Q9, R9, and S9. Q R S Dilate Triangle C on the coordinate plane using point (, 1) as the center of dilation and a scale factor of (,1) (,) C (,1) Dilate Triangle C on the coordinate plane using point (3, 3) as the center of dilation and a scale factor of (3,1) (3,3) C (15,3) M1-13 TOPIC : Similarit

15 5. Verif that each pair of triangles is similar. a. b. 1 1 (1,1) E (,) E (,9) (,) C (1,) (,5) D (,) C (,3) (,3) F (1,) D (,5) F (,5) c. d. 1 (,1) E (17,1) (1,11) (11,11) D (13,11) E (1,11) (,) C (,) F (15.5,) D (5,) F (17,) C (,1) Stretch Square CD has coordinates (, ), (, ), C (, ), and D (, ). dilation of Square CD has coordinates 9 (, ), 9 (, ), C9 (, ), and D9 (, ). What is the center of dilation? LESSON : Rising, Running, Stepping, Scaling M1-139

16 Review 1. Triangle XYZ has been enlarged with P as the center of dilation to form Triangle X Y Z. Identif the equivalent ratios that are equal to the scale factor. X' X P Y Z Y' Z'. triangle is dilated with center of dilation at point U. Point E is a verte of the triangle, and point E is the corresponding verte of the image. If UE 5 centimeters and UE 5 centimeters, what is the scale factor? 3. The coordinates of Quadrilateral CD are (, ), ( 5, 3), C (7, 3), and D (, ). What are the coordinates of the image if the quadrilateral is translated units right and 3 units down?. The coordinates of DJKL are J (, 1), K (, ), and L (, ). What are the coordinates of the image if the triangle is translated units left? 5. Write two unit rates for each situation. a. Julie can deliver 1 1 of the newspapers in hour. b. It took the author 3 of the ear to write 1 of the book. M1-1 TOPIC : Similarit