Kaleidoscopes The vibrant images seen in a kaleidoscope are produced by compositions of reflections in intersecting mirrors. Determine the measure of the angle between the mirrors in each kaleidoscope image.

Identify each mapping as a translation, reflection, rotation, or glide reflection. Find the translation rule, reflection line, center of rotation and angle of rotation, or glide translation rule and reflection line.

Image Long Description

40. Δ A B C → Δ E D C cap delta eh b c rightwards arrow cap delta e d c
41. Δ E D C → Δ P Q M cap delta e d c rightwards arrow cap delta p q m
42. Δ M N J → Δ E D C cap delta m n j rightwards arrow cap delta e d c
43. Δ H I F → Δ H G F cap delta h i f rightwards arrow cap delta h g f
44. Δ P Q M → Δ J L M cap delta p q m rightwards arrow cap delta j l m
45. Δ M N P → Δ E D C cap delta m n p rightwards arrow cap delta e d c
46. Δ J L M → Δ M N J cap delta j l m rightwards arrow cap delta m n j
47. Δ P Q M → Δ K J N cap delta p q m rightwards arrow cap delta k j n
48. Δ K J N → Δ A B C cap delta k j n rightwards arrow cap delta eh b c
49. Δ H G F → Δ K J N cap delta h g f rightwards arrow cap delta k j n

P → P ′ ( 3 , − 1 ) p rightwards arrow , p prime , open 3 comma negative 1 close  for the given translation and reflection line. Find the coordinates of P.

50. ( x , y ) → ( x − 3 , y ) ; y = 2 open x comma y close rightwards arrow open x minus 3 comma y close semicolon y equals 2
51. ( x , y ) → ( x , y − 3 ) ; y = 2 open x comma y close rightwards arrow open x comma y minus 3 close semicolon y equals 2
52. ( x , y ) → ( x − 3 , y − 3 ) ; y = x open x comma y close rightwards arrow open x minus 3 comma y minus 3 close semicolon y equals x
53. ( x , y ) → ( x + 4 , y − 4 ) ; y = − x open x comma y close rightwards arrow open x plus 4 comma y minus 4 close semicolon y equals negative x

C Challenge

54. Describe a glide reflection that maps the black R to the blue R.

    

For the given transformation mapping X Y ¯ x y bar  to X ′ Y ′ ¯ , x prime , y prime bar . comma  prove that X Y ¯ ≅ X ′ Y ′ ¯ . x y bar , approximately equal to . x prime , y prime bar . .

55. a translation
56. a reflection
57. a rotation
58. Reasoning Does an x ° x degrees  rotation about a point P followed by a reflection across a line ℓ script l  give the same image as a reflection across ℓ script l  followed by an x ° x degrees  rotation about P? Explain.

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