Reflections. Essential Question How can you reflect a figure in a coordinate plane?

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1 11. Reflections ssential Question How can ou reflect a figure in a coordinate plane? Reflecting a Triangle Using a Reflective evice Work with a partner. Use a straightedge to draw an triangle on paper. Label it. a. Use the straightedge to draw a line that does not pass through the triangle. Label it. b. Place a reflective device on line. c. Use the reflective device to plot the iages of the vertices of. Label the iages of vertices,, and as,, and, respectivel. d. Use a straightedge to draw b connecting the vertices. LOOKING OR STRUTUR To be proficient in ath, ou need to look closel to discern a pattern or structure. Reflecting a Triangle in a oordinate Plane Work with a partner. Use dnaic geoetr software to draw an triangle and label it. a. Refl ect in the -ais to for. b. What is the relationship between the coordinates of the vertices of and those of? c. What do ou observe about the side lengths and angle easures of the two triangles? d. Refl ect in the -ais to for. Then repeat parts (b) and (c). Saple Points ( 3, 3) 3 (, 1) ( 1, ) Segents =.1 1 = = ngles 1 = = 5.35 = 5.13 ounicate Your nswer 3. How can ou reflect a figure in a coordinate plane? Section 11. Reflections 551

2 11. Lesson What You Will Learn ore Vocabular reflection, p. 55 line of reflection, p. 55 glide reflection, p. 55 line setr, p. 555 line of setr, p. 555 Perfor reflections. Perfor glide reflections. Identif lines of setr. Solve real-life probles involving reflections. Perforing Reflections ore oncept Reflections reflection is a transforation that uses a line like a irror to reflect a figure. The irror line is called the line of reflection. reflection in a line aps ever point P in the plane to a point P, so that for P each point one of the following properties P P is true. P If P is not on, then is the perpendicular bisector of PP, or If P is on, then P = P. point P not on point P on Reflecting in Horizontal and Vertical Lines Graph with vertices (1, 3), (5, ), and (, 1) and its iage after the reflection described. a. In the line n: = 3 b. In the line : = 1 a. Point is units left of line n, so its reflection is units right of line n at (5, 3). lso, is units left of line n at (1, ), and is 1 unit right of line n at (, 1). b. Point is units above line, so is units below line at (1, 1). lso, is 1 unit below line at (5, 0). ecause point is on line, ou know that =. n Help in nglish and Spanish at igideasmath.co Graph fro aple 1 and its iage after a reflection in the given line. 1. =. = 3 3. =. = 1 55 hapter 11 Transforations

3 RMMR The product of the slopes of perpendicular lines is 1. ecause the slope of = is 1 and 1( 1) = 1, the slope of is 1. Reflecting in the Line = Graph G with endpoints ( 1, ) and G(1, ) and its iage after a reflection in the line =. Graph G and the line =. The slope of = is 1. The segent fro to its iage,, is perpendicular to the line of reflection =, so the slope of will be 1. ro, ove 1.5 units right and 1.5 units down to =. ro that point, ove 1.5 units right and 1.5 units down to locate (, 1). The slope of GG will also be 1. ro G, ove 0.5 unit right and 0.5 unit down to =. Then ove 0.5 unit right and 0.5 unit down to locate G (, 1). G = G You can use coordinate rules to find the iages of points reflected in four special lines. ore oncept oordinate Rules for Reflections If (a, b) is reflected in the -ais, then its iage is the point (a, b). If (a, b) is reflected in the -ais, then its iage is the point ( a, b). If (a, b) is reflected in the line =, then its iage is the point (b, a). If (a, b) is reflected in the line =, then its iage is the point ( b, a). Reflecting in the Line = Graph G fro aple and its iage after a reflection in the line =. Graph G and the line =. Use the coordinate rule for reflecting in the line = to find the coordinates of the endpoints of the iage. Then graph the iage. (a, b) ( b, a) ( 1, ) (, 1) G(1, ) G (, 1) G G = The vertices of JKL are J(1, 3), K(, ), and L(3, 1). 5. Graph JKL and its iage after a reflection in the -ais.. Graph JKL and its iage after a reflection in the -ais. Help in nglish and Spanish at igideasmath.co 7. Graph JKL and its iage after a reflection in the line =. 8. Graph JKL and its iage after a reflection in the line =. 9. In aple 3, verif that is perpendicular to =. Section 11. Reflections 553

4 Perforing Glide Reflections Postulate Reflection Postulate reflection is a rigid otion. ecause a reflection is a rigid otion, and a rigid otion preserves length and angle easure, the following stateents are true for the reflection shown. =, =, = =, =, = ecause a reflection is a rigid otion, the oposition Theore guarantees that an coposition of reflections and translations is a rigid otion. STUY TIP The line of reflection ust be parallel to the direction of the translation to be a glide reflection. glide reflection is a transforation involving a translation followed b a reflection in which ever point P is apped to a point P b the following steps. Step 1 irst, a translation aps P to P. Step Then, a reflection in a line k parallel to the direction of the translation aps P to P. P P Q Q Q P k Perforing a Glide Reflection Graph with vertices (3, ), (, 3), and (7, 1) and its iage after the glide reflection. Translation: (, ) ( 1, ) Reflection: in the -ais egin b graphing. Then graph after a translation 1 units left. inall, graph after a reflection in the -ais. (, 3) (, 3) ( 9, ) ( 9, ) ( 5, 1) ( 5, 1) (, 3) (3, ) (7, 1) 8 Help in nglish and Spanish at igideasmath.co 10. WHT I? In aple, is translated units down and then reflected in the -ais. Graph and its iage after the glide reflection. 11. In aple, describe a glide reflection fro to. 55 hapter 11 Transforations

5 Identifing Lines of Setr figure in the plane has line setr when the figure can be apped onto itself b a reflection in a line. This line of reflection is a line of setr, such as line at the left. figure can have ore than one line of setr. Identifing Lines of Setr How an lines of setr does each heagon have? a. b. c. a. b. c. 1 eterine the nuber of lines of setr for the figure Help in nglish and Spanish at igideasmath.co 15. raw a heagon with no lines of setr. Solving Real-Life Probles inding a Miniu istance You are going to bu books. Your friend is going to bu s. Where should ou park to iniize the distance ou both will walk? Reflect in line to obtain. Then draw. Label the intersection of and as. ecause is the shortest distance between and and =, park at point to iniize the cobined distance, +, ou both have to walk. Help in nglish and Spanish at igideasmath.co 1. Look back at aple. nswer the question b using a reflection of point instead of point. Section 11. Reflections 555

6 11. ercises naic Solutions available at igideasmath.co Vocabular and ore oncept heck 1. VOULRY glide reflection is a cobination of which two transforations?. WHIH ON OSN T LONG? Which transforation does not belong with the other three? plain our reasoning. and Modeling with Matheatics In ercises 3, deterine whether the coordinate plane shows a reflection in the -ais, -ais, or neither In ercises 7 1, graph JKL and its iage after a reflection in the given line. (See aple 1.) 7. J(, ), K(3, 7), L(, 1); -ais 8. J(5, 3), K(1, ), L( 3, ); -ais 9. J(, 1), K(, 5), L(3, 1); = J(1, 1), K(3, 0), L(0, ); = 11. J(, ), K(, ), L( 1, 0); = 1 1. J(3, 5), K(, 1), L(0, 3); = 3 In ercises 13 1, graph the polgon and its iage after a reflection in the given line. (See aples and 3.) 13. = 1. = 15. = 1. = 55 hapter 11 Transforations

7 In ercises 17 0, graph RST with vertices R(, 1), S(7, 3), and T(, ) and its iage after the glide reflection. (See aple.) 17. Translation: (, ) (, 1) Reflection: in the -ais 7. MOLING WITH MTHMTIS You park at soe point K on line n. You deliver a pizza to House H, go back to our car, and deliver a pizza to House J. ssuing that ou can cut across both lawns, how can ou deterine the parking location K that iniizes the distance HK + KJ? (See aple.) 18. Translation: (, ) ( 3, ) Reflection: in the line = Translation: (, ) (, + ) J H Reflection: in the line = 3 n 0. Translation: (, ) ( +, + ) Reflection: in the line = 8. TTNING TO PRISION Use the nubers and In ercises 1, deterine the nuber of lines of setr for the figure. (See aple 5.) 1. sbols to create the glide reflection resulting in the iage shown.. ( 1, 5) (5, ) ( 1, 1) 3. (, ) (3, ). 8 Translation: (, ) Reflection: in = (, ) ( ), 5. USING STRUTUR Identif the line setr (if an) of each word. a. LOOK b. MOM c. OX d. 1. RROR NLYSIS escribe and correct the error in 8 + In ercises 9 3, find point on the -ais so + is a iniu. 30. (, 5), (1, 3) 31. ( 8, ), ( 1, 3) 8 3. ( 1, 7), (5, ) 33. MTHMTIL ONNTIONS The line = 3 + to is a glide reflection. is reflected in the line = 1. What is the equation of the iage? Section 11. int_ath1_pe_110.indd (1, ), (, 1) describing the transforation. Reflections 557 1/9/15 :39 PM

8 HOW O YOU S IT? Use igure. igure 35. ONSTRUTION ollow these steps to construct a reflection of in line. Use a copass and straightedge. Step 1 raw and line. Step Use one copass setting to find two points that are equidistant fro on line. Use the sae copass setting to find a point on the other side of that is the sae distance fro these two points. Label that point as. Step 3 Repeat Step to find points and. raw. igure 1 igure 3. USING TOOLS Use a reflective device to verif our construction in ercise MTHMTIL ONNTIONS Reflect MNQ in the line =. = M igure 3 igure a. Which figure is a reflection of igure in the line = a? plain. b. Which figure is a reflection of igure in the line = b? plain. c. Which figure is a reflection of igure in the line =? plain. d. Is there a figure that represents a glide reflection? plain our reasoning. Maintaining Matheatical Proficienc Use the diagra to find the angle easure. (Section 8.5) 0. O 1. O. O 3. O. O 5. O. O 7. O 8. O 9. O Q 5 N THOUGHT PROVOKING Is the coposition of a translation and a reflection coutative? (In other words, do ou obtain the sae iage regardless of the order in which ou perfor the transforations?) Justif our answer. 39. MTHMTIL ONNTIONS Point (1, ) is the iage of (3, ) after a reflection in line c. Write an equation for line c. Reviewing what ou learned in previous grades and lessons O hapter 11 Transforations