Worksheet: Combining Transformations

In this worksheet, we will practice carrying out and describing combinations of transformations.

Q1:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has then been transformed onto triangle 𝐴′′𝐡′′𝐢′′.

Describe the single transformation that maps 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • Aa rotation of 90∘ clockwise about the point 𝐷
  • Ba dilation from the point 𝐷 by a scale factor of 2
  • Ca dilation from the point 𝐴 by a scale factor of 3
  • Da rotation of 180∘ about the point 𝐷
  • Ea dilation from the point 𝐷 by a scale factor of 3

Describe the single transformation that maps 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • Aa rotation of 90∘ clockwise about the point 𝐷
  • Ba rotation of 180∘ about the point 𝐴
  • Ca rotation of 180∘ about the point 𝐷
  • Da rotation of 180∘ about the point 𝐡
  • Ea rotation of 90∘ counterclockwise about the point 𝐷

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ similar?

  • Ayes
  • Bno

Q2:

A shape 𝐹 has been reflected in the line 𝑦=π‘₯ and then rotated 270∘ clockwise about the origin to 𝐹′. Would 𝐹 and 𝐹′ be congruent?

  • Ayes
  • Bno

Q3:

In the given figure, triangle 𝐴 has been transformed to triangle 𝐡. Which of the following sequences of transformations could have been used?

  • Aa reflection in the line 𝑦=1 followed by a reflection in the π‘₯-axis
  • Ba 90∘ rotation about the point (βˆ’2,βˆ’2) followed by a translation three right
  • Ca 180∘ rotation about the point (βˆ’2,2) followed by a translation three right
  • Da 180∘ rotation about the origin followed by a translation three up
  • Ea reflection in the π‘₯-axis followed by a translation four right and four up

Q4:

In the given figure, triangle 𝐴 has been transformed to triangle 𝐴′ by reflecting first in the 𝑦-axis and then reflecting in the π‘₯ axis. What single transformation would have mapped 𝐴 to 𝐴′?

  • Aa rotation about the origin of 90∘
  • Ba rotation about the origin of 270∘
  • Ca rotation about the origin of 180∘
  • Da reflection in the 𝑦-axis
  • Ea reflection in the line 𝑦=π‘₯

Q5:

𝐴 𝐡 𝐢 𝐷 is reflected in the π‘₯-axis and then translated 5 units to the right. What is the image of point 𝐡?

  • A ( βˆ’ 2 , βˆ’ 6 )
  • B ( βˆ’ 6 , βˆ’ 2 )
  • C ( 2 , 6 )
  • D ( βˆ’ 6 , βˆ’ 7 )
  • E ( βˆ’ 7 , βˆ’ 6 )

Q6:

Starting with triangle 𝐴(3,7), 𝐡(4,1), and 𝐢(8,7), apply the transformations: 1. reflect in the 𝑦-axis, 2. reflect in the π‘₯-axis, and 3. translate by 3 units right and 3 units up. What are the images of the vertices?

  • A 𝐴 β€² ( 6 , βˆ’ 4 ) , 𝐡 β€² ( 7 , 2 ) , 𝐢 β€² ( βˆ’ 5 , βˆ’ 4 )
  • B 𝐴 β€² ( 0 , βˆ’ 4 ) , 𝐡 β€² ( βˆ’ 1 , 2 ) , 𝐢 β€² ( βˆ’ 5 , βˆ’ 4 )
  • C 𝐴 β€² ( 0 , βˆ’ 4 ) , 𝐡 β€² ( βˆ’ 1 , 2 ) , 𝐢 β€² ( βˆ’ 8 , βˆ’ 7 )
  • D 𝐴 β€² ( 0 , βˆ’ 4 ) , 𝐡 β€² ( βˆ’ 1 , 4 ) , 𝐢 β€² ( βˆ’ 5 , 1 0 )
  • E 𝐴 β€² ( 6 , 1 0 ) , 𝐡 β€² ( βˆ’ 1 , 2 ) , 𝐢 β€² ( βˆ’ 5 , 1 0 )

Q7:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has then been transformed onto triangle 𝐴′′𝐡′′𝐢′′ then transformed onto 𝐴′′′𝐡′′′𝐢′′′ as seen in the figure.

Describe the single transformation that would map 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • Aa 90∘ clockwise rotation about 𝐸
  • Ba 90∘ clockwise rotation about 𝐷
  • Ca 90∘ counterclockwise rotation about 𝐷
  • Da 90∘ counterclockwise rotation about 𝐸
  • Ea translation one left and three up

Describe the single transformation that would map 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • Aa 90∘ clockwise rotation about 𝐹
  • Ba 90∘ clockwise rotation about 𝐸
  • Ca 180∘ rotation about 𝐹
  • Da reflection in the line ⃖⃗𝐸𝐹
  • Ea translation two up

Describe the single transformation that would map 𝐴′′𝐡′′𝐢′′ onto 𝐴′′′𝐡′′′𝐢′′′.

  • Aa translation three right and two down
  • Ba translation two right and three down
  • Ca translation two left and three up
  • Da translation three left and two up
  • Ea translation three left and two down

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′′𝐡′′′𝐢′′′ congruent?

  • Ayes
  • Bno

Q8:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has then been transformed onto triangle 𝐴′′𝐡′′𝐢′′.

Describe the single transformation that maps 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • Aa translation two up
  • Ba 90∘ rotation counterclockwise about point 𝐸
  • Ca 90∘ rotation clockwise about point 𝐸
  • Da translation two down
  • Ea reflection in 𝐷𝐸

Describe the single transformation that maps 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • Aa dilation from the point 𝐷 by a scale factor of 2
  • Ba dilation from the point 𝐸 by a scale factor of 12
  • Ca translation two up
  • Da dilation from the point 𝐸 by a scale factor of 2
  • Ea translation two down

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ similar?

  • Ayes
  • Bno

Q9:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has then been transformed onto triangle 𝐴′′𝐡′′𝐢′′.

Describe the single transformation that maps 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • AA 90∘ counterclockwise rotation about the origin
  • BA 90∘ clockwise rotation about the origin
  • CA dilation from the origin by a scale factor of 2
  • DA 270∘ clockwise rotation about the origin
  • EA dilation from point (0,3) by a scale factor of 2

Describe the single transformation that maps 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • AA dilation from point (0,3) by a scale factor of 2
  • BA dilation from the origin by a scale factor of 2
  • CA dilation from the origin by a scale factor of 3
  • DA 180∘ clockwise rotation about point (0,6)
  • EA 180∘ clockwise rotation about point (0,5)

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ similar?

  • AYes
  • BNo

Q10:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has then been transformed onto triangle 𝐴′′𝐡′′𝐢′′.

Describe the single transformation that maps 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • Aa translation two right and three up
  • Ba translation three right and one up
  • Ca reflection in the 𝑦-axis
  • Da translation three up and one right
  • Ea reflection in the π‘₯-axis

Describe the single transformation that maps 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • Aa translation one left and two up
  • Ba dilation from the origin by a scale factor of 2
  • Ca dilation from point (0,2) by a scale factor of 12
  • Da dilation from point (0,4) by a scale factor of 12
  • Ea translation one right and two up

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ similar?

  • Ayes
  • Bno

Q11:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴𝐡𝐢 which has then been transformed onto triangle 𝐴𝐡𝐢 as seen in the figure.

Describe the single transformation that would map 𝐴𝐡𝐢 to 𝐴′𝐡′𝐢′.

  • Aa reflection in the line ⃖⃗𝐷𝐸
  • Ba rotation of 180∘ about 𝐸
  • Ca rotation of 180∘ about 𝐷
  • Da rotation of 90∘ clockwise about point 𝐸
  • Ea rotation of 90∘ clockwise about point 𝐷

Describe the single transformation that would map 𝐴′𝐡′𝐢′ to 𝐴′′𝐡′′𝐢′′.

  • Aa rotation of 90∘ clockwise about point 𝐸
  • Ba rotation of 90∘ counterclockwise about point 𝐸
  • Ca rotation of 90∘ clockwise about point 𝐹
  • Da rotation of 90∘ counterclockwise about point 𝐹
  • Ea translation two right

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ congruent?

  • Ayes
  • Bno

Q12:

The triangles 𝐴𝐡𝐢 and 𝐴′𝐡′𝐢′in the figure are similar. Which of the following statements justifies this?

  • ATriangle 𝐴𝐡𝐢 can be mapped onto triangle 𝐴′𝐡′𝐢′ by a sequence of transformations: first, a reflection in the ⃖⃗𝐸𝐹, and then a dilation of the image by a scale factor of three from point 𝐷.
  • BTriangle 𝐴𝐡𝐢 can be mapped onto triangle 𝐴′𝐡′𝐢′ by a sequence of transformations: first, a dilation by a scale factor of three from point 𝐷, and then a translation of the image two down.
  • CTriangle 𝐴𝐡𝐢 can be mapped onto triangle 𝐴′𝐡′𝐢′ by a sequence of transformations: first, a dilation by a scale factor of three from point 𝐷, and then a translation of the image eight down.
  • DTriangle 𝐴𝐡𝐢 can be mapped onto triangle 𝐴′𝐡′𝐢′ by a sequence of transformations: first, a dilation by a scale factor of three from point 𝐷, and then a reflection of the image in the ⃖⃗𝐸𝐹.
  • ETriangle 𝐴𝐡𝐢 can be mapped onto triangle 𝐴′𝐡′𝐢′ by a sequence of transformations: first, a reflection in the ⃖⃗𝐸𝐹, and then a translation of the image four right.

Q13:

Does there exist a series of similarity transformations that would map quadrilateral 𝐴𝐡𝐢𝐷 to quadrilateral 𝐻𝐺𝐹𝐸? If yes, explain your answer.

  • AThere is no series of similarity transformations.
  • BYes, quadrilateral 𝐴𝐡𝐢𝐷 could be dilated by a scale factor of 3, rotated, and then reflected to quadrilateral 𝐻𝐺𝐹𝐸.
  • CYes, quadrilateral 𝐴𝐡𝐢𝐷 could be dilated by a scale factor of 2 and then rotated to quadrilateral 𝐻𝐺𝐹𝐸.
  • DYes, quadrilateral 𝐴𝐡𝐢𝐷 could be dilated by a scale factor of 3 and then reflected to quadrilateral 𝐻𝐺𝐹𝐸.
  • EYes, quadrilateral 𝐴𝐡𝐢𝐷 could be dilated by a scale factor of 2, rotated, and then reflected to quadrilateral 𝐻𝐺𝐹𝐸.

Q14:

The triangle with vertices (3,3), (7,0), and (10,5) was transformed to (1,8), (5,5), and (8,10) and then to (1,8), (βˆ’2,4), and (3,1). Which of the following describes these transformations?

  • AIt was translated 2 units right and 5 units down, and then it was rotated 90∘counterclockwise about the point (1,8).
  • BIt was rotated 180∘counterclockwise about the point (3,3), and then it was translated 2 units right and 5 units down.
  • CIt was translated 2 units left and 5 units up, and then it was rotated 180∘clockwise about the point (1,8).
  • DIt was translated 2 units left and 5 units up, and then it was rotated 90∘clockwise about the point (1,8).

Q15:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has then been transformed onto triangle 𝐴′′𝐡′′𝐢′′ as seen in the figure.

Describe the single transformation that would map 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • Aa 90∘ counterclockwise rotation about point 𝐷
  • Ba 90∘ clockwise rotation about point 𝐷
  • Ca 180∘ rotation about point 𝐷
  • Da translation four left
  • Ea translation two right and eight down

Describe the single transformation that would map 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • Aa translation two right and eight down
  • Ba translation two left and eight up
  • Ca translation eight right and two down
  • Da translation two left and eight down
  • Ea translation two right and eight up

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ congruent?

  • Ano
  • Byes

Q16:

The quadrilateral 𝐴𝐡𝐢𝐷 has been transformed onto quadrilateral 𝐴𝐡𝐢𝐷 which has then been transformed onto quadrilateral 𝐴𝐡𝐢𝐷.

Describe the single transformation that maps 𝐴𝐡𝐢𝐷 onto 𝐴′𝐡′𝐢′𝐷′.

  • Aa reflection in the π‘₯-axis
  • Ba dilation from the point (βˆ’2,0) by a scale factor of 3
  • Ca translation two right
  • Da reflection in the 𝑦-axis
  • Ea translation two left

Describe the single transformation that maps 𝐴′𝐡′𝐢′𝐷′ onto 𝐴′′𝐡′′𝐢′′𝐷′′.

  • Aa reflection in the 𝑦-axis
  • Ba dilation from the point (βˆ’2,0) by a scale factor of 13
  • Ca dilation from the point (βˆ’2,0) by a scale factor of 3
  • Da dilation from the point (0,βˆ’2) by a scale factor of 3
  • Ea reflection in the π‘₯-axis

Hence, are quadrilaterals 𝐴𝐡𝐢𝐷 and 𝐴′′𝐡′′𝐢′′𝐷′′ similar?

  • Ano
  • Byes

Q17:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴𝐡𝐢 which has then been transformed onto triangle 𝐴𝐡𝐢.

Describe the single transformation that maps 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • Aa dilation from the point 𝐡 by a scale factor of 3
  • Ba dilation from the point 𝐷 by a scale factor of 3
  • Ca translation two right and two up
  • Da translation two left and two down
  • Ea dilation from the point 𝐷 by a scale factor of 2

Describe the single transformation that maps 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • Aa translation six right
  • Ba 90∘ counterclockwise rotation about the point 𝐷
  • Ca 180∘ counterclockwise rotation about the point 𝐷
  • Da translation six left
  • Ea 90∘ rotation counterclockwise about the point 𝐷

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ similar?

  • Ano
  • Byes

Q18:

Find the image of point (βˆ’10,βˆ’9) after applying the translation (π‘₯,𝑦)β†’(π‘₯βˆ’8,𝑦+5) followed by a 90∘ rotation counterclockwise about the origin.

  • A ( 4 , βˆ’ 1 8 )
  • B ( βˆ’ 1 8 , 4 )
  • C ( 9 , βˆ’ 8 )
  • D ( βˆ’ 1 4 , 2 )
  • E ( βˆ’ 4 , βˆ’ 1 8 )

Q19:

Reflect Triangle 𝐡 in the 𝑦-axis and then in the π‘₯-axis. Which triangle is its image?

  • A 𝐴
  • B 𝐡
  • C 𝐢
  • D 𝐷

Q20:

Reflect Triangle 𝐴 in the 𝑦-axis and then in the π‘₯-axis. Which triangle is its image?

  • A 𝐷
  • B 𝐢
  • C 𝐡
  • D 𝐴

Q21:

In the given figure, what combination of transformations would map circle 𝐴 onto circle 𝐡?

  • Aa translation of four left and six down, followed by a dilation of scale factor 12
  • Ba translation of six left and four down, followed by a dilation of scale factor 23
  • Ca translation of four left and six down, followed by a dilation of scale factor 23
  • Da translation of six right and four up, followed by a dilation of scale factor 32
  • Ea translation of four left and six up, followed by a dilation of scale factor 34

Q22:

Does there exist a series of similarity transformations that would map triangle 𝐴𝐡𝐢 to triangle 𝐸𝐹𝐷? If yes, explain your answer.

  • ANo series of similarities exists because the two triangles are of different sizes.
  • BYes, triangle 𝐴𝐡𝐢 could be dilated by a scale factor of 3, rotated, and then reflected.
  • CYes, triangle could be dilated by a scale factor of 3 and then reflected.
  • DYes, triangle 𝐴𝐡𝐢 could be dilated by a scale factor of 2 and then reflected.
  • EYes, triangle 𝐴𝐡𝐢 could be dilated by a scale factor of 2 and then rotated.

Q23:

In the given figure, what combination of transformations would map circle 𝐴 onto circle 𝐡?

  • Aa translation of two right and six down followed by a dilation of scale factor two
  • Ba translation of two left and six up followed by a dilation of scale factor two
  • Ca translation of four left and two down followed by a dilation of scale factor one
  • Da translation of six left and two up followed by a dilation of scale factor one
  • Ea translation of six right and two down followed by a dilation of scale factor two

Q24:

First, translate the given triangle two right and two down, and then rotate this image 180∘ about the origin. Which of the following sets of coordinates will be the vertices of the image?

  • A ( 1 , 0 ) , ( 1 , 1 ) , and (βˆ’2,1)
  • B ( 0 , βˆ’ 1 ) , ( βˆ’ 1 , βˆ’ 1 ) , and (βˆ’1,2)
  • C ( 0 , 1 ) , ( 1 , 1 ) , and (1,2)
  • D ( 0 , 1 ) , ( 1 , 1 ) , and (1,βˆ’2)
  • E ( βˆ’ 1 , 0 ) , ( 1 , 1 ) , and (2,1)

Q25:

The triangle 𝐴𝐡𝐢 has been transformed to triangle 𝐴𝐡𝐢 which has then been transformed to triangle 𝐴𝐡𝐢.

Describe the single transformation that maps 𝐴𝐡𝐢 to 𝐴′𝐡′𝐢′.

  • Aa dilation from the origin by a scale factor of 2
  • Ba dilation from the origin by a scale factor of 3
  • Ca reflection in the π‘₯-axis
  • Da dilation from point (0,1) by a scale factor of 3
  • Ea reflection in the 𝑦-axis

Describe the single transformation that maps 𝐴′𝐡′𝐢′ to 𝐴′′𝐡′′𝐢′′.

  • Aa dilation from the origin by a scale factor of 3
  • Ba reflection in the 𝑦-axis
  • Ca dilation from the origin by a scale factor of 13
  • Da dilation from the origin by a scale factor of 2
  • Ea reflection in the π‘₯-axis

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ similar?

  • Ayes
  • Bno

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