TRANSFORMATIONS AND SYMMETRY

Transcription

1 TRNSFORMTIONS ND SYMMETRY Studing transforations of geoetric shapes buids a foundation for a ke idea in geoetr: congruence. In this introduction to transforations, the students epore three rigid otions: transations, refections, and rotations. These eporations are done with tracing paper as we as with dnaic toos on the coputer or other device. Students app one or ore of these otions to the origina shape, creating its iage in a new position without changing its size or shape. Rigid transforations aso ead direct to studing setr in shapes. These ideas wi hep with describing and cassifing geoetric shapes ater in the chapter. See the Math Notes boes in Lessons and for ore inforation about rigid transforations. Eape 1 Decide which rigid transforation was used on each pair of shapes beow. Soe a be a cobination of transforations. a. b. c. d. e. f. Identifing a singe transforation is usua eas for students. In part (a), the paraeogra is refected (fipped) across an invisibe vertica ine. (Iagine a irror running vertica between the two figures. One figure woud be the refection of the other.) Refecting a shape once changes its orientation. For eape, in part (a), the two sides of the figure at eft sant upwards to the right, whereas in its refection at right, the sant upwards to the eft. Likewise, the anges in the figure at eft switch positions in the figure at right. In part (b), the shape is transated (or sid) to the right and down. The orientation reains the sae, with a sides santing the sae. Parent Guide with Etra Practice 5

2 Part (c) shows a cobination of transforations. First the triange is refected (fipped) across an invisibe horizonta ine. Then it is transated (sid) to the right. The pentagon in part (d) has been rotated (turned) cockwise to create the second figure. Iagine tracing the first figure on tracing paper, then hoding the tracing paper with a pin at one point beow the first pentagon, then turning the paper to the right 90. The second pentagon woud be the resut. Soe students ight see this as a refection across a diagona ine. The pentagon itsef coud be, but with the added dot (sa circe), the entire shape cannot be a refection. If it had been refected, the dot woud have to be on the corner beow the one shown in the rotated figure. The trianges in part (e) are rotations of each other (90 again). Part (f) shows another cobination. The triange is rotated (the shortest side becoes horizonta instead of vertica) and refected. Eape 2 What wi the figure at right ook ike if it is first refected across ine and then the resut is refected across ine? P The first refection is the new figure shown between the two ines. If we were to join each verte (corner) of the origina figure to its corresponding verte on the second figure, those ine segents woud be perpendicuar to ine and the vertices of (and a the other points in) the refection woud be the sae distance awa fro as the are in the origina figure. One wa to draw the refection is to use tracing paper to trace the figure and the ine. Then turn the tracing paper over, so that ine is on top of itsef. This wi show the position of the refection. Transfer the figure to our paper b tracing it. Repeat this process with ine to for the third figure b tracing. P s students discovered in cass, refecting twice ike this across two intersecting ines produces a rotation of the figure about the point P. Put the tracing paper back over the origina figure to ine. Put a pin or the point of a pen or penci on the tracing paper at point P (the intersection of the ines of refection) and rotate the tracing paper unti the origina figure wi fit perfect on top of the ast figure. P 6 ore onnections Geoetr

3 Eape 3 The shape at right is trapezoid D. Transate the trapezoid 7 units to the right and 4 units up. Labe the new trapezoid D and give the coordinates of its vertices. Is it possibe to transate the origina trapezoid in such a wa to create D so that it is a refection of D? If so, what woud be the refecting ine? Wi this awas be possibe for an figure? ( 5, 2) ( 3, 2) D( 6, 4) ( 2, 4) Transating (or siding) the trapezoid 7 units to the right and 4 units up gives a new trapezoid (2, 2), (4, 2), (5, 0), and D (1, 0). If we go back to trapezoid D, we now wonder if we can transate it in such a wa that we can ake it ook as if it were a refection rather than a transation. Since the trapezoid is setrica, it is possibe to do so. We can side the trapezoid horizonta eft or right. In either case, the resuting figure woud ook ike a refection. This wi not awas work. It works here because we started with an isoscees trapezoid, which has a ine of setr itsef. Students epored which pogons have ines of setr, and which have rotationa setr as we. gain, the used tracing paper as we as technoog to investigate these properties. ( 5, 2) ( 3, 2) D( 6, 4) ( 2, 4) D (1, 0) (2, 2) (4, 2) (5, 0) Eporing these transforations and setrica properties of shapes heps to iprove students visuaization skis. These skis are often negected or taken for granted, but uch of atheatics requires students to visuaize pictures, probes, or situations. That is wh we ask students to visuaize or iagine what soething ight ook ike as we as practice creating transforations of figures. Parent Guide with Etra Practice 7

4 Probes Perfor the indicated transforation on each pogon beow to create a new figure. You a want to use tracing paper to see how the figure oves. 1. Rotate Figure 90 cockwise 2. Refect Figure across ine. about the origin. 3. Transate Figure 6 units eft. 4. Rotate Figure D 270 cockwise about the origin (0, 0). D For probes 5 through 20, refer to the figures beow. Figure Figure Figure 8 ore onnections Geoetr

5 State the new coordinates after each transforation. 5. Transate Figure eft 2 units and down 3 units. 6. Transate Figure right 3 units and down 5 units. 7. Transate Figure eft 1 unit and up 2 units. 8. Refect Figure across the -ais. 9. Refect Figure across the -ais. 10. Refect Figure across the -ais. 11. Refect Figure across the -ais. 12. Refect Figure across the -ais. 13. Refect Figure across the -ais. 14. Rotate Figure 90 countercockwise about the origin. 15. Rotate Figure 90 countercockwise about the origin. 16. Rotate Figure 90 countercockwise about the origin. 17. Rotate Figure 180 countercockwise about the origin. 18. Rotate Figure 180 countercockwise about the origin. 19. Rotate Figure 270 countercockwise about the origin. 20. Rotate Figure 90 cockwise about the origin. 21. Pot the points (3, 3), (6, 1), and (3, 4). Transate the triange 8 units to the eft and 1 unit up to create Δ. What are the coordinates of the new triange? 22. How can ou transate Δ in the ast probe to put point at (4, 5)? 23. Refect Figure Z across ine, and then refect the new figure across ine. What are these two refections equivaent to? Z Parent Guide with Etra Practice 9

6 For each shape beow, (i) draw a ines of setr, and (ii) describe its rotationa setr if it eists nswers D D 10 ore onnections Geoetr

7 5. ( 1, 3) (1, 2) (3, 1) 6. ( 2, 3) (2, 3) (3, 0) 7. ( 5, 4) (3, 4) ( 3, 1) 8. (1, 0) (3, 4) (5, 2) 9. ( 5, 2) ( 1, 2) (0, 5) 10. ( 4, 2) (4, 2) ( 2, 3) 11. ( 1, 0) ( 3, 4) ( 5, 2) 12. (5, 2) (1, 2) (0, 5) 13. (4, 2) ( 4, 2) (2, 3) 14. (0, 1) ( 4, 3) ( 2, 5) 15. ( 2, 5) ( 5, 0) ( 2, 1) 16. ( 2, 4) ( 2, 4) (3, 2) 17. ( 1, 0) ( 3, 4) ( 5, 2) 18. (4, 2) ( 4, 2) (2, 3) 19. (2, 5) (2, 1) (5, 0) 20. (2, 4) (2, 4) ( 3, 2) 21. (5, 4), ( 2, 2), ( 5, 3) 22. Transate it 1 unit right and 8 units down. Z 23. The two refections are the sae as rotating Z about point X. Z X 24. This has 180 rotationa setr. Z 25. The one ine of setr. No rotationa setr. 26. The circe has infinite an ines of setr, ever one of the iustrates refection setr. It aso has rotationa setr for ever possibe degree easure. 27. This irreguar shape has no ines of setr and does not have rotationa setr, nor refection setr. Parent Guide with Etra Practice 11