Understanding Isometries, Rotations, Translations, and Compositions

2. Isometries-100 home answer What is the definition of an isometry? Give three examples of isometries.

3. Isometries-100Answer home question Isometry: a transformation that perseveres length, angle measure, parallel lines, etc. ex. Reflections Rotations Translations

4. Isometries-200 answer home Which of the following is not a rotation of ? c) a) b)

5. Isometries-200Answer home question c)

6. Isometries-300 home answer True or false? 1) Transformations that are not isometries are called rigid transformations. 2) Flips, turns and slides are nicknames for reflections, rotations and translations 3) Isometries preserve angle measures and parallel lines

7. Isometries-300Answer home question • False • True • True

8. Isometries-400 home answer Find the value of each variable if the given transformation is an isometry 50° 4d 7 a° 50° b° 12 2c+3

9. Isometries-400Answer home question a=90° c=2 2c+3=7 2c=4 c=2 b=40° d=3 180-90-50 4d=12 90-50 d=3 40

10. Isometries-500 home answer Is the given transformation an isometry? ABC XYZ A=(-4,2) X=(2,2) B=(-1,4) Y=(4,-1) C=(-1,1) Z=(1,-1)

11. Isometries-500Answer question home Yes Use the distance formula to compare the side lengths AB=√(-4+1)²+(2-4)² BC=√(-1+1)²+(4-1)² AC=√(-4+1)²+(2-1)² =√(-3)²+(-2)² =√(0)²+(3)² =√(-3)²+(1)² =√9+4 =√9 =√9+1 =√13 =3 =√10 XY=√(2-4)²+(2+1)² YZ=√(4-1)²+(-1+1)² XZ=√(2-1)²+(2+1)² =√(-2)²+(3) ² =√(3)²+(0)² =√(1)²+(3)² =√4+9 =√9 =√1+9 =√13 =3 =√10 AB=XY BC=YZ AC=XZ

12. Rotations-100 home answer Does this figure have rotational symmetry? If so, describe the rotation that maps the figure onto itself.

13. Rotations-100Answer home question Yes, the star does have rotational symmetry. To map the figure onto itself, you could rotate the object 72° or 144°.

14. Rotations-200 home answer A=(2,-3) Al=(-3,-2) If A was rotated clockwise around the origin, what was the angle of rotation?

15. Rotations-200Answer home question 90° In a 90° clockwise rotation, (x,y) (y,-x) If you use that information, you can substitute in (2,-3) to get (-2,-3), which are the coordinates of the given pre-image and image

16. Rotations-300 home answer m K What is the measure of the angle of rotation? .Al .A .All 138°

17. Rotations-300Answer home question 84 ° When you reflect a figure over line k then over line m, the angle of rotation is 2x (x=the measure of the acute angle formed by k and m) So, x=180-138 x=42 2(42)=84°

18. Rotations-400 home answer Rotate (7,-2) 90°clockwise around the origin. Name the point of the image. Do the same for 180° and 270° clockwise.

19. Rotations-400Answer home question 90°=(-2, -7) because (x,y) (y,-x) 180°=(-7,2) because (x,y) (-x,-y) 270°=(2,7) because (x,y) (-y,x)

20. Rotations-500 answer home 5c Find the values of all the variables 4b 10 65° a° 8 5 2d+2

21. Rotations-500Answer home question a=130° c=1 a=2(65) 5c=5 a=130 c=1 b=2 d=4 4b=8 2d+2=10 b=2 2d=8 d=4

22. Translations-100 answer home Reflect AB, A=(3,-3) B= (2,-4), over y=1. What are the coordinates of Al and Bl

23. Translations-100Answer question home Al=(3,5) Bl=(2,-6)

24. Translation-200 home answer Find the other endpoint using the following vectors. 1.(-4,0) vector <2,-3> 2. (5, -2) vector <5,1>

25. Translation-200Answer home question • (-2,-3) • (-4+2,0-3) • (-2,-3) • 2. (10,-1) • (5+5,-2+1) • (10,-1)

26. Translation-300 answer home • Use the following coordinate notation to find the other endpoint. • (x, y) (x+2, y-3) • (1,4) • 2. (-3, -1)

27. Translation-300Answer question home • (3,1) (1+2,4-3) (3,1) • (-1,-4) (-3+2,-1-3) (-1,-4)

28. Translation-400 home answer A translation of AB is described by vector PQ<2,-5>. Find the value of each variable. A(w-5,-3)Al(10,x-1) B(z,3y+1)Bl(5,5)

29. question Translation-400Answer home w=10 y=3 w-5+2=10 3y+1-5=5 w-3=10 3y-4=5 w=13 3y=9 x=-7 y=3 -3-5=x-1 z=3 -8=x-1 z+2=5 -7=x z=3

30. Translation-500 home answer Write the equation for the line of reflection A=(2,3) B=(6,-1)

31. Translation-500Answer home question y= x-3 Explanation: (2,3) (6,-1) Slope= (3+1)=-1 midpoint=(6+2 3-1)= (4,1) (2-6) 2 , 2 Perp. Line slope=1 y=1x+b 1=1(4)+b 1=4+b -3=b y=x-3

32. Compositions-100 home answer What is a composition? What is a glide reflection?

33. Compositions-100Answer question home A composition is when 2 or more transformations are combined to form a single transformation A glide reflection is a transformation in which every point P is mapped onto Pll by the following 2 steps -a translation that maps P onto Pl -a reflection in line k such that the line of translation is parallel to reflection line k

34. Compositions-200 home answer When you switch the order of transformations, does it affect the final image? In what cases?

35. Compostitions-200Answer question home In a composition, it does affect the final image, but it does not in a glide reflection.

36. Compostitions-300 home answer Rotate A(3,2) 90° about the origin and reflect over the x-axis.

37. Compositions-300Answer home question Al (2,-3) All(2,3)

38. Compositions-400 answer home Sketch the image of AB, A(4,2) B(7,0), after a composition using the given transformations (in the given order) Translation: (x,y) (x-4,y+2) Rotation: 270° clockwise about the origin

39. Compositions-400 Answer question home Translation: A(0,4) B(3,2) Rotation: A(-4,0) B(-2,3)

40. Compositions-500 answer home Sketch the image of A (-5,2) after translating it using vector <3,-4> and reflecting over x=4

41. Compositions-500Answer home question After translation: Al(-2,-2) After reflection: All(10,-2)