Graphing a Reflection Image

Transcription

1 9- Reflections oon ore State Standards G-O.. Given a geoetric figure and a rotation, reflection, or translation, draw the transfored figure.... lso G-O.., G-O.., G-O..6 MP 1, MP 3, MP Objective To find reflection iages of figures If it s hard to visualize, tr using grid paper to draw the shapes. Look at the shapes at the right. Visualize flipping each shape across its ellow line. What word do the iages of the shapes for? op the shapes as the are shown and sketch the results of flipping the. MTHEMTIL PRTIES In the Solve It, ou reflected shapes across lines. Notice that when ou reflect a figure, the shapes have opposite orientations. Two figures have opposite orientations if the corresponding vertices of the preiage and iage read in opposite directions. Lesson Vocabular reflection line of reflection The vertices of UG read clockwise. G U U G The vertices of U G read counterclockwise. Essential Understanding When ou reflect a figure across a line, each point of the figure aps to another point the sae distance fro the line but on the other side. The orientation of the figure reverses. Ke oncept Reflection cross a Line reflection across a line, called the line of reflection, is a transforation with the following properties: If a point is on line, then the iage of is itself (that is, = ). If a point is not on line, then is the perpendicular bisector of. You write the reflecion across that takes P to P as R (P) = P. The preiage and its iage are equidistant fro the line of reflection. hapter 9 Transforations

2 You can use the equation of a line of reflection in the function notation. For eaple, R = describes the reflection across the line =. Proble 1 Reflecting a Point cross a Line Multiple hoice Point P has coordinates (3, ). What are the coordinates of R =1 (P)? (3, -) (0, ) (3, -) (-3, -) Graph point P and the line of reflection = 1. P and its reflection iage across the line ust be equidistant fro the line of reflection. How does a graph help ou visualize the proble? graph shows that = 1 is a horizontal line, so the line through P that is perpendicular to the line of reflection is a vertical line. P 1 O P Move along the line through P that is perpendicular to the line of reflection. Stop when the distances of P and P to the line of reflection are the sae. P is 3 units above the line = 1, so P ust be 3 units below the line = 1. The line = 1 is the perpendicular bisector of PP if P is (3, -). The correct answer is. 1. R =1 (P) = P. What are the coordinates of P? You can also use the notation R to describe reflections of figures. The diagra below shows R ( ), and function notation is used to describe soe of the properties of reflections. Propert Properties of Reflections Reflections preserve distance. If R () =, and R () =, then =. Reflections preserve angle easure. If R ( ) =, then =. Reflections ap each point of the preiage to one and onl one corresponding point of its iage. R () = if and onl if R ( ) =. Observe that the above properties ean that reflections are rigid otions, which ou learned about in Lesson 9-1. Lesson 9- Reflections

3 Proble Graphing a Reflection Iage oordinate Geoetr Graph points ( 3, ), (0, 1), and (, ). Graph and label ( ). Step 1 Graph. Show the -ais as the dashed line of reflection. Step Find,, and. is in the sae position as because is on the line of reflection. Locate and so that the -ais is the perpendicular bisector of and. Step 3 raw. intersects the line of reflection. How will the iage relate to the line of reflection? The iage will also intersect the line of reflection. O O O. Graph fro Proble. Graph and label ( ). If Triangle is the iage of a reflection, what do ou know about the preiage? The preiage has opposite orientation, and lies on the opposite side of the line of reflection. Proble 3 Writing a Reflection Rule Each triangle in the diagra is a reflection of another triangle across one of the given lines. How can ou describe Triangle b using a reflection rule? Triangle is the iage of a reflection, so find the preiage and the line of reflection to write a rule. The preiage cannot be Triangle 3 because Triangle and Triangle 3 have the sae orientation and reflections reverse orientation. heck Triangles 1 and b drawing line segents that connect the corresponding vertices of Triangle. ecause neither line k nor line is the perpendicular bisector of the segent drawn fro Triangle 1 to Triangle, Triangle 1 is not the preiage. Line k is the perpendicular bisector of the segents joining corresponding vertices of Triangle and Triangle. So, Triangle = R k (Triangle ). 1 3 k How can ou use a reflection rule to describe Triangle 1? Eplain. k 6 hapter 9 Transforations

4 You can use the properties of reflections to prove stateents about figures. Proble Using Properties of Reflections What do ou have to know about GHJ to show that it is an isosceles triangle? Isosceles triangles have at least two congruent sides. In the diagra, R t (G) = G, R t (H) = J, and R t () =. Use the properties of reflections to describe how ou know that GHJ is an isosceles triangle. Since R t (G) = G, R t (H) = J, and reflections preserve distance, R t (GH ) = GJ. So, GH = GJ and, b definition, GHJ is an isosceles triangle. H G J. an ou use properties of reflections to prove that GHJ is equilateral? Eplain. t Lesson heck o ou know HOW? Use the graph of FGH. 1. What are the coordinates of R - ais (H)?. What are the coordinates of R =3 (G)? 3. Graph and label R = ( FGH). F G O H o ou UNERSTN? MTHEMTIL PRTIES. Vocabular What is the relationship between a line of reflection and a segent joining corresponding points of the preiage and iage?. Error nalsis classate sketched R s () = as shown in the diagra. a. Eplain our classate s error. b. op point and line s and show the correct location of. s 6. What are the coordinates of a point P(, ) reflected across the -ais? cross the -ais? Use reflection notation to write our answer. Practice and Proble-Solving Eercises MTHEMTIL PRTIES Practice Find the coordinates of each iage. See Proble R =1 (Q) 8. R = -1 (P) P S 9. R - ais (S) 10. R =0. (T) V O 11. R = -3 (U) 1. R - ais (V ) T Q U Lesson 9- Reflections 7

5 oordinate Geoetr Given points J(1, ), (3, ), and G(, 1), graph JG and its reflection iage as indicated. See Proble. 13. R - ais 1. R - ais 1. R = 16. R = 17. R = R = 19. Each figure in the diagra at the right is a reflection of another figure across one of the reflection lines. a. Write a reflection rule to describe Figure 3. Justif our answer. b. Write a reflection rule to describe Figure. Justif our answer. Figure 1 c. Write a reflection rule to describe Figure. Justif our answer. 0. In the diagra at the right, LMNP is a rectangle with LM = MN. a. op the diagra. Then sketch R LM (LMNP). b. What figure results fro the reflection? Use properties of reflections to justif our solution. j n L P Figure 3 Figure Figure M N See Proble 3. See Proble. ppl op each figure and line O. raw each figure s reflection iage across line O oordinate Geoetr The following steps eplain how to reflect point across the line =. Step 1 raw line / through (, 1) perpendicular to the line =. The slope of = is 1, so the slope of line / is 1 # (-1), or -1. Step Fro, ove two units left and two units up to =. Then ove two ore units left and two ore units up to find the O location of on line /. The coordinates of are (1, ). a. op the diagra. Then draw the lines through and that are perpendicular to the line =. What is the slope of each line? b. R = () = and R = () =. What are the coordinates of and? c. Graph. d. Make a onjecture opare the coordinates of the vertices of and. Make a conjecture about the coordinates of the point P(a, b) reflected across the line =.. oordinate Geoetr has vertices (-3, ), (-, -1), and (0, 3). Graph R =- ( ) and label it. (Hint: See Eercise 3.) 8 hapter 9 Transforations

6 Q. Recreation When ou pla pool, ou can use the fact that P the ball bounces off the side of the pool table at the sae angle at which it hits the side. Suppose ou want to put the ball at point into the pocket at point P b bouncing it off side RS. Off what point on RS should the ball bounce? raw a diagra and eplain our reasoning. 6. Think bout a Plan The coordinates of the vertices of S FGH are F(, -1), G(-, -), and H(-, 3). Graph FGH and R =-3 ( FGH). What is the relationship between the line = - 3 and FF, GG, and HH? How can ou use slope to find the iage of each verte? R 7. In the diagra R(E) = E. a. What are the idpoints of and? b. What is the equation of the line of reflection? c. Write a rule that describes this reflection. op each pair of figures. Then draw the line of reflection ou can use to ap one figure onto the other E E 30. Histor The work of artist and scientist Leonardo da Vinci (1 119) has an unusual characteristic. His handwriting is a irror iage of noral handwriting. a. Write the irror iage of the sentence, Leonardo da Vinci was lefthanded. Use a irror to check how well ou did. b. Eplain wh the fact about da Vinci in part (a) ight have ade irror writing see natural to hi. 31. Open-Ended Give three eaples fro everda life of objects or situations that show or use reflections. Find the iage of O(0, 0) after two reflections, first across line O 1 and then across line O. 3. / 1 : = 3, / : -ais 33. / 1 : = -, / : -ais 3. / 1 : -ais, / : -ais 3. Reasoning When ou reflect a figure across a line, does ever point on the preiage ove the sae distance? Eplain. 36. Securit Recall that when a ra of light hits a irror, it bounces off the irror at the sae angle at which it hits the irror. You are installing a securit caera. t what point on the irrored wall should ou ai the caera at in order to view the door at? raw a diagra and eplain our reasoning. Mirrored wall Lesson 9- Reflections 9

7 hallenge 37. Use the diagra at the right. Find the coordinates of each iage point. a. R = () = b. R =- ( ) = c. R = ( ) = d. R =- ( ) = e. How are and related? (1, 3) Reasoning an ou for the given tpe of quadrilateral b drawing a triangle and then reflecting one or ore ties? Eplain. 38. parallelogra 39. isosceles trapezoid 0. kite 1. rhobus. rectangle 3. square. oordinate Geoetr Show that R = () = for point (a, b) and (b, a). (Hint: Show that = is the perpendicular bisector of.) PERFORMNE TSK ppl What You ve Learned Look back at the inforation about the video gae on page 3, and review the requireents for ase 1 and ase of the video gae progra. The graph of the puzzle piece and target area is shown again below. Select all of the following that are true. Eplain our reasoning. MTHEMTIL PRTIES MP 1. T <, - > (, ) oves so that the iage of is the reflection of F across the -ais.. T <, - > (, ) oves so that the iage of is the reflection of EF across the -ais.. R = () = F. F is the reflection of (- 1, 1) across the line =. O F E E. F is the reflection of across the line = +. F. E is the reflection of across the line = +. G. The iage of the puzzle piece after a reflection across a line has the sae orientation as the target area. H. The proble cannot be solved for ase 1 using onl reflections. 60 hapter 9 Transforations