2 Notation:Pre-Image: original figure Image: after transformation. Use prime notationA’CC ’BB’A
3 Isometry AKA: congruence transformation a transformation in which an original figure and its image are congruent.
4 Theorems about isometries FUNDAMENTAL THEOREM OF ISOMETRIESAny any two congruent figures in a plane can be mapped onto one another by at most 3 reflectionsISOMETRY CLASSIFICATION THEOREMThere are only 4 isometries. They are:
5 TRANSLATION: moves all points in a plane a given direction a fixed distance
6 TRANSLATION VECTOR: Direction Magnitude PRE-IMAGEIMAGE
18 PARTNER SWAP: Part I: (Live under my rules) Use sketchpad to graph & label any three pointsGraph & Reflect them over the line y = xGraph->Plot new function->x->OKConstruct two points on the line and connect themMark this line segment as your mirror.WRITE a conjecture about how (a, b) will be changed after reflecting over y = x. Explain.Repeat by reflecting over the line y = -x. Write a conjecture.
19 Starter:Find one vector which would accomplish the same thing as translating (3, -1) by <3, 8> then applying the transformation T(x, y)->(x-4, y+9)Find coordinates of (7, 6) reflected over:a.) the y-axisb.) the x-axisc.) the line y = xd.) the line x = -33. HW Check & Peer edit
20 Rotations have:Center of rotationAngle of rotation:
21 Example: Rotate Triangle ABC 60 degrees clockwise about “its center” Find the image of A after a 120 degree rotationFind the image of A after a 180 degree rotationFind the image of A after a 240 degree rotationFind the image of A after a 300 degree rotationFind the image of A after a 360 degree rotation
23 ROTATIONS PRESERVE SIZE SHAPE ORIENTATION Length of sides Measure of anglesAreaPerimeterSHAPEORIENTATION
24 PARTNER SWAP: Part II: (Live under new rules) Use sketchpad to graph & label any three points. Connect them and construct triangle interior.Rotate your pre-image about the origin 90Rotate the pre-image about the origin 180Rotate the pre-image about the origin 270Rotate the pre-image about the origin 360WRITE A CONJECTURE: What are the coordinates of (a, b) after a 90, 180, and 270 degree rotation about the origin?
25 Rotations on a coordinate plane about the origin 90 (a, b) -> (-b, a)180 (a, b) -> (-a, -b)270 (a, b) -> (b, -a)360 (a, b) -> (a, b)
26 DEBRIEFING: Find the coordinates of (2, 5) Reflected over the x-axis Reflected over the y-axisReflected over the line x = 3Reflected over the line y = -2Reflected over the line y = xRotated about the origin 180Rotated about the origin 270 Rotated about the origin 360
27 Review the rules for coordinate geometry transformations Which two transformations would accomplish the same thing as a 90 degree rotation about the origin?Use sketchpad to justify your answer
28 Coordinate Geometry rules Reflectionsx axis (a, b) -> (a, -b)y axis (a, b) -> (-a, b)y=x (a, b) -> (b, a)Rotations about the origin90 (a, b) -> (-b, a)180 (a, b) -> (-a, -b)270 (a, b) -> (b, -a)360 (a, b) -> (a, b)
29 GLIDE REFLECTIONS You can combine different Geometric Transformations…
30 Practice: Reflect over y = x then translate by the vector <2, -3>
33 Santucci’s Starter:Complete the following transformations on (6, 1) and list coordinates of the image:a. Reflect over the x-axisb. Reflect over the y-axisc. Rotate 90 about the origind. Rotate 180 about the origine. Rotate 270 about the originEXPLAIN in writing: what two transformations would accomplish the same thing as a 90 degree rotation about the origin?
34 Starter:Find the coordinates of pre-image (3, 4) after the following transformations (do without graphing…)reflect over y-axisreflect over x-axisreflect over y=xreflect over y=-xtranslate <-2, 6>rotate 90 about originrotate 180 about originrotate 270 about originrotate 360 about origin
35 PAIRS Sketchpad Exploration: Rotate (3, 4) 90 degrees about the point (1, 6). What two transformations will produce the same result?Try it again by rotating (3, 4) 90 degrees about (-2, 5).Rotate (2, -6) 90 degrees about (1, 7)Describe OR LIST STEPS FOR how you can find the image of any point after a 90 rotation about (a, b).Try it again with a 180 rotation about (a,b). How can you find the image?Try it again with a 270 rotation about (a,b). How can you find the image?
36 Starter HW Peer edit Practice 12-5 Reflectional symmetryBoth rotational and Reflectional symmetrySee keyNo lines of symmetryLine symmetry (5 lines) and 72 degree rotational symmetryLine symmetry (1 line)Line symmetry (4 lines) and 90 degree rotational symmetryLine symmetry (8 lines) and 45 degree rotational symmetry180 degree rotational symmetryLine symmetry (1 line) #17-21 see key
37 Symmetry Line Symmetry Rotational Symmetry If a figure can be reflected onto itself over a line.Rotational SymmetryIf a figure can be rotated about some point onto itself through a rotation between 0 and 360 degrees
38 What kinds of symmetry do each of the following have?
39 What kinds of symmetry do each of the following have? Rotational (180) Point SymmetryRotational (90, 180, 270)Point SymmetryRotational (60, 120, 180, 240, 300)Point Symmetry
40 NOTE: TEST WILL BE END OF NEXT WEEK!!! Isometry Wrap Up…Sketchpad Activitiy # 6 Symmetry in Regular PolygonsDilations ExplorationNOTE: TEST WILL BE END OF NEXT WEEK!!!
41 DilationsPlot any 5 points to make a convex polygon and fill in its interior red.Mark the origin as center.Make the polygon larger by a scale factor of 2 and fill it in green.Make the polygon smaller by a scale factor of 1/3. Fill it in red.Measure your coordinates and Explain how you can find coordinates of a dilation image.Try marking a new center and dilating a few points. What is the “center” of a dilation? How does it change the measurements?
42 Tessellations web-quest VISIT:Explore & read information underTessellations:What are theyThe beginningsSymmetry & MC EscherThe galleriesSolid StuffAnswer the following questions:1. What is symmetry and list the types discussed.2. What are the Polya’ symmetries?3. How many Polya’ symmetries are there?4. What are the Rhomboid possibilities?5. What is the difference between a periodic and aperiodic tiling?
43 TO-DO Complete Tessellations Sketchpad explorations, # 8, 9 Read rubric and write questions. Begin design
44 INDIRECT PROOF If ~q then ~p Assume that the conclusion is FALSE.Reason to a contradiction.If n>6 then the regular polygon will not tessellate.ASSUME: The polygon tessellatesSHOW: n can not be >6
45 Indirect proof Regular polygons with n>6 sides will not tessellate Assume a polygon with n>6 sides will tessellate.This means that n*one interior <measure will equal 360IF n = 3 there are 6 angles about center pointIF n = 4 there are 4 angles about center pointIF n = 6 there are 3 angles about center pointTherefore, if n>6 then there must be fewer than 3 angles about the center point. In other words, there must be 2 or fewer. If there are 2 angles about the center point then each angle must measure 180 to sum to 360But no regular polygon exists whose interior angle measures 180 (int. < sum must be LESS than 180). Therefore, the polygon can not tessellate.
46 Santucci’s StarterDetermine if the following will tessellate & provide proof:Isosceles triangleKiteRegular pentagonRegular hexagonRegular heptagonRegular octagonRegular nonagonRegular decagon
47 Review practiceFind the image of A(-1, 4) reflected over the x-axis then over the y-axis (two intersecting lines). What one transformation would accomplish the same result?Find the image of B(6, -2) reflected over x=3 then over x=-5 (two parallel lines). What one transformation would accomplish the same result?List all the rotational symmetries of a regular decagon.Draw a regular octagon with all its lines of symmetry (on sketchpad).
49 Coordinate Transformations MOAT game Groups of “3”Write answer on white board and send one “runner” to stand facing the class with representatives from all other groups (hold board face down). When MOAT is called flip answer so all members seated can see answer.1st group correct = +3 points2nd group correct = +2 points3rd group correct = +1 pointsGroup with HIGHEST # points +3 on quizGroup with 2nd highest # points +2 on quizGroup with 3rd highest # points +1 on quiz
50 HW Answers p. 650 12-4 10. H 4. F translate twice the distance 11. M 13. Segment BC14. A15. Segment LM16. I17. K34. a.) B(-2, 5)b.) C(-5, -2)c.) D(2, -5)d.) Square: 4 congruent sides & angles12-44. F translate twice the distanceTranslate T across m twice the distance between l and mV rotated 145Peer editopp; reflectionsame; translationsame; 270 rotationGlide <-2, -2>, reflect over y = x – 128. Glide <0, 4>, reflect over y = 0 (x-axis)