Essential Question What conjectures can you make about a figure reflected in two lines?

Transcription

1 OO O earning tandard -O..5 -O..6. OTUTI VI UT To be proficient in ath, ou need to ae conjectures and justif our conclusions. ongruence and Transforations ssential uestion What conjectures can ou ae about a figure reflected in two lines? eflections in arallel ines Wor with a partner. Use dnaic geoetr software to draw an scalene triangle and label it. a. raw an line. eflect aple in to for. b. raw a line parallel to. eflect in the new line to for. c. raw the line through point that is perpendicular to. What do ou notice? d. ind the distance between points and. ind the distance between the two parallel lines. What do ou notice? e. ide. Is there a single transforation that aps to? xplain. f. ae conjectures based on our answers in parts (c) (e). Test our conjectures b changing and the parallel lines. eflections in Intersecting ines Wor with a partner. Use dnaic geoetr software to draw an scalene triangle and label it. a. raw an line. eflect aple in to for. b. raw an line so that angle is less than or equal to 90. eflect in to for. c. ind the easure of. otate counterclocwise about point using an angle twice the easure of. d. ae a conjecture about a figure reflected in two intersecting lines. Test our conjecture b changing and the lines. ounicate Your nswer 3. What conjectures can ou ae about a figure reflected in two lines?. oint is reflected in two parallel lines, and, to for and. The distance fro to is 3. inches. What is the distance? ection. ongruence and Transforations 199

2 . esson What You Will earn ore Vocabular congruent figures, p. 00 congruence transforation, p. 01 Identif congruent figures. escribe congruence transforations. Use theores about congruence transforations. Identifing ongruent igures Two geoetric figures are congruent figures if and onl if there is a rigid otion or a coposition of rigid otions that aps one of the figures onto the other. ongruent figures have the sae size and shape. ongruent ot congruent sae size and shape different sizes or shapes You can identif congruent figures in the coordinate plane b identifing the rigid otion or coposition of rigid otions that aps one of the figures onto the other. ecall fro ostulates.1.3 and Theore.1 that translations, reflections, rotations, and copositions of these transforations are rigid otions. Identifing ongruent igures Identif an congruent figures in the coordinate plane. xplain. I 5 OUTIO quare is a translation of square units left and 6 units down. o, square and square are congruent. 5 x is a reflection of in the x-axis. o, and are congruent. U TU is a 180 rotation of I. o, I and TU are congruent. 5 T onitoring rogress elp in nglish and panish at igideasath.co 1. Identif an congruent figures in the coordinate plane. xplain. I T x U 00 hapter Transforations

3 I You can read the notation as parallelogra,,,. ongruence Transforations nother nae for a rigid otion or a cobination of rigid otions is a congruence transforation because the preiage and iage are congruent. The ters rigid otion and congruence transforation are interchangeable. escribe a congruence transforation that aps to. escribing a ongruence Transforation x OUTIO The two vertical sides of rise fro left to right, and the two vertical sides of fall fro left to right. If ou reflect in the -axis, as shown, then the iage,, will have the sae orientation as. Then ou can ap to using a translation of units down. x o, a congruence transforation that aps to is a reflection in the -axis followed b a translation of units down. onitoring rogress elp in nglish and panish at igideasath.co. In xaple, describe another congruence transforation that aps to. 3. escribe a congruence transforation that aps to. x ection. ongruence and Transforations 01

4 Using Theores about ongruence Transforations opositions of two reflections result in either a translation or a rotation. coposition of two reflections in parallel lines results in a translation, as described in the following theore. Theore Theore. eflections in arallel ines Theore If lines and are parallel, then a reflection in line followed b a reflection in line is the sae as a translation. If is the iage of, then 1. is perpendicular to and, and. = d, where d is the distance between and. roof x. 31, p. 06 d Using the eflections in arallel ines Theore In the diagra, a reflection in line aps to. reflection in line aps to. lso, = 9 and =. a. ae an segents congruent to each segent:,, and. b. oes =? xplain. c. What is the length of? OUTIO a., and... b. Yes, = because and are perpendicular to both and. o, and are opposite sides of a rectangle. c. the properties of reflections, = 9 and =. The eflections in arallel ines Theore iplies that = =, so the length of is (9 + ) = 6 units. onitoring rogress elp in nglish and panish at igideasath.co Use the figure. The distance between line and line is 1.6 centieters.. The preiage is reflected in line, then in line. escribe a single transforation that aps the blue figure to the green figure. 5. What is the relationship between and line? xplain. 6. What is the distance between and? 0 hapter Transforations

5 coposition of two reflections in intersecting lines results in a rotation, as described in the following theore. Theore Theore.3 eflections in Intersecting ines Theore If lines and intersect at point, then a reflection in line followed b a reflection in line is the sae as a rotation about point. The angle of rotation is x, where x is the easure of the acute or right angle fored b lines and. x x roof x. 31, p. 50 = x Using the eflections in Intersecting ines Theore In the diagra, the figure is reflected in line. The iage is then reflected in line. escribe a single transforation that aps to. 70 OUTIO the eflections in Intersecting ines Theore, a reflection in line followed b a reflection in line is the sae as a rotation about point. The easure of the acute angle fored between lines and is 70. o, b the eflections in Intersecting ines Theore, the angle of rotation is (70 ) = 10. single transforation that aps to is a 10 rotation about point. You can chec that this is correct b tracing lines and and point, then rotating the point 10. onitoring rogress 7. In the diagra, the preiage is reflected in line, then in line. escribe a single transforation that aps the blue figure onto the green figure. 8. rotation of 76 aps to. To ap to using two reflections, what is the easure of the angle fored b the intersecting lines of reflection? elp in nglish and panish at igideasath.co 80 ection. ongruence and Transforations 03

6 . xercises naic olutions available at igideasath.co Vocabular and ore oncept hec 1. OT T T Two geoetric figures are if and onl if there is a rigid otion or a coposition of rigid otions that oves one of the figures onto the other.. VOUY Wh is the ter congruence transforation used to refer to a rigid otion? onitoring rogress and odeling with atheatics In xercises 3 and, identif an congruent figures in the coordinate plane. xplain. (ee xaple 1.) 3.. U x T V 3 T U V x 6. 6 W Z X Y 6 x In xercises 7 10, deterine whether the polgons with the given vertices are congruent. Use transforations to explain our reasoning. 7. (, ), (5, ), (, 1) and T(6, ), U(9, ), V(8, 1) 8. W( 3, 1), X(, 1), Y(, ), Z( 5, ) and ( 1, 3), ( 1, ), (, ), (, 5) 9. (1, 1), (3, ), (, 1) and (6, 1), (5, ), (, 1) 10. (0, 0), (1, ), (, ), (3, 0) and (0, 5), ( 1, 3), (, 3), ( 3, 5) In xercises 5 and 6, describe a congruence transforation that aps the blue preiage to the green iage. (ee xaple.) 5. 6 x In xercises 11 1,, is reflected in line, and is reflected in line. (ee xaple 3.) 11. translation aps onto which triangle? 1. Which lines are perpendicular to? 13. If the distance between and is.6 inches, what is the length of? 1. Is the distance fro to the sae as the distance fro to? xplain. 0 hapter Transforations

7 In xercises 15 and 16, find the angle of rotation that aps onto. (ee xaple.) 15. In xercises 19, find the easure of the acute or right angle fored b intersecting lines so that can be apped to using two reflections. 19. rotation of 8 aps to. 0. rotation of aps to The rotation (x, ) ( x, ) aps to.. The rotation (x, ) (, x) aps to OI Use the eflections in arallel ines Theore (Theore.) to explain how ou can ae a glide reflection using three reflections. ow are the lines of reflection related?. WI OUIO The pattern shown is called a tessellation. 17. O YI escribe and correct the error in describing the congruence transforation. x a. What transforations did the artist use when creating this tessellation? b. re the individual figures in the tessellation congruent? xplain our reasoning. is apped to b a translation 3 units down and a reflection in the -axis. 18. O YI escribe and correct the error in using the eflections in Intersecting ines Theore (Theore.3). 7 ITI TII In xercises 5 8, tell whether the stateent is alwas, soeties, or never true. xplain our reasoning. 5. congruence transforation changes the size of a figure. 6. If two figures are congruent, then there is a rigid otion or a coposition of rigid otions that aps one figure onto the other. 7. The coposition of two reflections results in the sae iage as a rotation. 8. translation results in the sae iage as the coposition of two reflections. 7 rotation about point aps the blue iage to the green iage. 9. OI uring a presentation, a areting representative uses a projector so everone in the auditoriu can view the advertiseent. Is this projection a congruence transforation? xplain our reasoning. ection. ongruence and Transforations 05

8 30. OW O YOU IT? What tpe of congruence transforation can be used to verif each stateent about the stained glass window? a. Triangle 5 is congruent to Triangle 8. b. Triangle 1 is congruent to Triangle. c. Triangle is congruent to Triangle 7. d. entagon 3 is congruent to entagon OVI TO rove the eflections in arallel ines Theore (Theore.). d 7 iven reflection in line aps to, a reflection in line aps to, and. rove a. is perpendicular to and. b. = d, where d is the distance between and. 33. I UT, with endpoints (1, 3) and (3, ), is reflected in the -axis. The iage is then reflected in the x-axis to produce the iage. One classate sas that is apped to b the translation (x, ) (x, 5). nother classate sas that is apped to b a ( 90), or 180, rotation about the origin. Which classate is correct? xplain our reasoning. 3. ITI TII oes the order of reflections for a coposition of two reflections in parallel lines atter? or exaple, is reflecting XYZ in line and then its iage in line the sae as reflecting XYZ in line and then its iage in line? Y X Z OTUTIO In xercises 35 and 36, cop the figure. Then use a copass and straightedge to construct two lines of reflection that produce a coposition of reflections resulting in the sae iage as the given transforation. 35. Translation: 36. otation about : XYZ X Y Z 3. TOUT OVOI tessellation is the covering of a plane with congruent figures so that there are no gaps or overlaps (see xercise ). raw a tessellation that involves two or ore tpes of transforations. escribe the transforations that are used to create the tessellation. Z X Y Y Z X aintaining atheatical roficienc olve the equation. hec our solution. (ills eview andboo) eviewing what ou learned in previous grades and lessons 37. 5x + 16 = 3x = 39. b + 8 = 6b 0. 7w 9 = 13 w 1. 7(n + 11) = n. (8 ) = 6 3. ast ear, the trac tea s ard sale earned $500. This ear, the ard sale earned $65. What is the percent of increase? (ills eview andboo) 06 hapter Transforations