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Lesson: Combining Transformations

In this lesson, we will learn how to identify an image after a series of transformations and identify a transformation that is the same as a series of transformations.

Sample Question Videos

03:14

Worksheet • 21 Questions • 1 Video

Q1:

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

  • A 𝐷
  • B 𝐡
  • C 𝐴
  • D 𝐢

Q2:

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

  • A 𝐢
  • B 𝐴
  • C 𝐡
  • D 𝐷

Q3:

In the given figure, triangle 𝐴 has been transformed to triangle 𝐴 β€² by reflecting first in the 𝑦 -axis and then reflecting in the π‘₯ axis. What single transformation would have mapped 𝐴 to 𝐴 β€² ?

  • A a rotation about the origin of 1 8 0 ∘
  • Ba rotation about the origin of 2 7 0 ∘
  • C a reflection in the line 𝑦 = π‘₯
  • D a reflection in the 𝑦 -axis
  • E a rotation about the origin of 9 0 ∘

Q4:

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 ( βˆ’ 7 , βˆ’ 6 )
  • D ( 2 , 6 )
  • E ( βˆ’ 6 , βˆ’ 7 )

Q5:

𝐴 𝐡 𝐢 𝐷 is reflected in the 𝑦 -axis and then translated 2 units to the right. What is the image of point 𝐷 ?

  • A ( βˆ’ 2 , βˆ’ 5 )
  • B ( βˆ’ 5 , βˆ’ 2 )
  • C ( βˆ’ 4 , βˆ’ 5 )
  • D ( 2 , 5 )
  • E ( βˆ’ 5 , βˆ’ 4 )

Q6:

Does there exist a series of similarity transformations that would map quadrilateral 𝐴 𝐡 𝐢 𝐷 to quadrilateral 𝐻 𝐺 𝐹 𝐸 ? If yes, explain your answer.

  • A Yes, quadrilateral 𝐴 𝐡 𝐢 𝐷 could be dilated by a scale factor of 2, rotated, and then reflected to quadrilateral 𝐻 𝐺 𝐹 𝐸 .
  • BThere is no series of similarity transformations.
  • C Yes, quadrilateral 𝐴 𝐡 𝐢 𝐷 could be dilated by a scale factor of 3, rotated, and then reflected to quadrilateral 𝐻 𝐺 𝐹 𝐸 .
  • D Yes, quadrilateral 𝐴 𝐡 𝐢 𝐷 could be dilated by a scale factor of 3 and then reflected to quadrilateral 𝐻 𝐺 𝐹 𝐸 .
  • E Yes, quadrilateral 𝐴 𝐡 𝐢 𝐷 could be dilated by a scale factor of 2 and then rotated to quadrilateral 𝐻 𝐺 𝐹 𝐸 .

Q7:

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

  • Aa translation of six left and four down, followed by a dilation of scale factor 2 3
  • Ba translation of four left and six up, followed by a dilation of scale factor 3 4
  • Ca translation of four left and six down, followed by a dilation of scale factor 2 3
  • Da translation of four left and six down, followed by a dilation of scale factor 1 2
  • Ea translation of six right and four up, followed by a dilation of scale factor 1 3

Q8:

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

  • A a translation of six right and two down followed by a dilation of scale factor two
  • B a translation of four left and two down followed by a dilation of scale factor one
  • C a translation of two right and six down followed by a dilation of scale factor two
  • D a translation of two left and six up followed by a dilation of scale factor two
  • E a translation of six left and two up followed by a dilation of scale factor one

Q9:

Does there exist a series of similarity transformations that would map triangle 𝐴 𝐡 𝐢 to triangle 𝐸 𝐹 𝐷 ? If yes, explain your answer.

  • A Yes, triangle 𝐴 𝐡 𝐢 could be dilated by a scale factor of 3, rotated, and then reflected.
  • B No series of similarities exists because the two triangles are of different sizes.
  • C Yes, triangle 𝐴 𝐡 𝐢 could be dilated by a scale factor of 2 and then reflected.
  • D Yes, triangle 𝐴 𝐡 𝐢 could be dilated by a scale factor of 2 and then rotated.
  • E Yes, triangle could be dilated by a scale factor of 3 and then reflected.

Q10:

Find the image of point ( βˆ’ 1 0 , βˆ’ 9 ) after translation by ( π‘₯ , 𝑦 ) β†’ ( π‘₯ βˆ’ 8 , 𝑦 + 5 ) followed by a rotation about the origin through an angle of 9 0 ∘ .

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

Q11:

Find the image of point ( βˆ’ 6 , βˆ’ 4 ) after translation by ( π‘₯ , 𝑦 ) β†’ ( π‘₯ + 8 , 𝑦 βˆ’ 4 ) followed by a rotation about the origin through an angle of 9 0 ∘ .

  • A ( 8 , 2 )
  • B ( 4 , 8 )
  • C ( 2 , 8 )
  • D ( 0 , 1 4 )
  • E ( βˆ’ 8 , 2 )

Q12:

Find the image of point ( 9 , 1 ) after translation by ( π‘₯ , 𝑦 ) β†’ ( π‘₯ , 𝑦 βˆ’ 4 ) followed by a rotation about the origin through an angle of 9 0 ∘ .

  • A ( 3 , 9 )
  • B ( βˆ’ 1 , 0 )
  • C ( 9 , 3 )
  • D ( 5 , βˆ’ 9 )
  • E ( βˆ’ 3 , 9 )

Q13:

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

  • AIt was translated 2 units left and 5 units up, and then it was rotated 9 0 ∘ clockwise about the point ( 1 , 8 ) .
  • BIt was translated 2 units right and 5 units down, and then it was rotated 9 0 ∘ counterclockwise about the point ( 1 , 8 ) .
  • CIt was rotated 1 8 0 ∘ counterclockwise about the point ( 3 , 3 ) , and then it was translated 2 units right and 5 units down.
  • DIt was translated 2 units left and 5 units up, and then it was rotated 1 8 0 ∘ clockwise about the point ( 1 , 8 ) .

Q14:

The triangle with vertices ( 3 , 1 ) , ( 1 , 4 ) , and ( 0 , 0 ) was transformed to ( 8 , βˆ’ 1 ) , ( 6 , 2 ) , and ( 5 , βˆ’ 2 ) and then to ( 8 , βˆ’ 1 ) , ( 1 0 , βˆ’ 4 ) , and ( 1 1 , 0 ) . Which of the following describes these transformations?

  • AIt was translated 5 units right and 2 units down, and then it was rotated 1 8 0 ∘ counterclockwise about the point ( 8 , βˆ’ 1 ) .
  • BIt was translated 5 units left and 2 units up, and then it was rotated 1 8 0 ∘ clockwise about the point ( 8 , βˆ’ 1 ) .
  • CIt was rotated 9 0 ∘ clockwise about the point ( 3 , 1 ) , and then it was translated 5 units left and 2 units up.
  • DIt was translated 5 units right and 2 units down, and then it was rotated 9 0 ∘ counterclockwise about the point ( 8 , βˆ’ 1 ) .

Q15:

A transformation maps point 𝐴 to point 𝐡 . We say that 𝐡 is the of 𝐴 .

  • Apreimage
  • Bimage
  • Cpostimage

Q16:

A transformation maps point 𝐴 to point 𝐡 . We say that 𝐴 is the of 𝐡 .

  • Apreimage
  • Bfigure
  • Cimage
  • Dorigin
  • Epostimage

Q17:

Triangle 𝐴 𝐡 𝐢 can be reflected and then translated onto 𝐷 𝐸 𝐹 .

Determine the length of 𝐡 𝐢 .

Determine the measure of angle 𝐷 𝐹 𝐸 .

Q18:

Which of the following statements will be true for two triangles that are similar.

  • AOne triangle can always be mapped to the other by a translation, reflection, rotation, or a combination of these transformations.
  • BOne triangle can always be mapped to the other by a translation, reflection, rotation, dilation, or a combination of these transformations.
  • COne triangle can sometimes (but not always) be mapped to the other by a translation, reflection, rotation, dilation, or a combination of these transformations.

Q19:

The triangles 𝐴 𝐡 𝐢 and 𝐴 β€² 𝐡 β€² 𝐢 β€² in the figure are similar. Which of the following statements justifies this?

  • A Triangle 𝐴 𝐡 𝐢 can be mapped onto triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² 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 βƒ–     βƒ— 𝐸 𝐹 .
  • B Triangle 𝐴 𝐡 𝐢 can be mapped onto triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² 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.
  • C Triangle 𝐴 𝐡 𝐢 can be mapped onto triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² by a sequence of transformations: first, a reflection in the βƒ–     βƒ— 𝐸 𝐹 , and then a translation of the image four right.
  • D Triangle 𝐴 𝐡 𝐢 can be mapped onto triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² by a sequence of transformations: first, a reflection in the βƒ–     βƒ— 𝐸 𝐹 , and then a dilation of the image by a scale factor of three from point 𝐷 .
  • E Triangle 𝐴 𝐡 𝐢 can be mapped onto triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² 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.

Q20:

The quadrilateral 𝐴 𝐡 𝐢 𝐷 has been transformed onto quadrilateral 𝐴 𝐡 𝐢 𝐷 β€² β€² β€² β€² which has then been transformed onto quadrilateral 𝐴 𝐡 𝐢 𝐷 β€² β€² β€² β€² β€² β€² β€² β€² .

Describe the single transformation that maps 𝐴 𝐡 𝐢 𝐷 onto 𝐴 β€² 𝐡 β€² 𝐢 β€² 𝐷 β€² .

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

Describe the single transformation that maps 𝐴 β€² 𝐡 β€² 𝐢 β€² 𝐷 β€² onto 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² 𝐷 β€² β€² .

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

Hence, are quadrilaterals 𝐴 𝐡 𝐢 𝐷 and 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² 𝐷 β€² β€² similar?

  • Ano
  • Byes

Q21:

The triangle 𝐴 𝐡 𝐢 has been transformed onto triangle 𝐴 𝐡 𝐢 β€² β€² β€² which has then been transformed onto triangle 𝐴 𝐡 𝐢 β€² β€² β€² β€² β€² β€² .

Describe the single transformation that maps 𝐴 𝐡 𝐢 onto 𝐴 β€² 𝐡 β€² 𝐢 β€² .

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

Describe the single transformation that maps 𝐴 β€² 𝐡 β€² 𝐢 β€² onto 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² .

  • Aa 9 0 ∘ rotation counterclockwise about the point 𝐷
  • Ba translation six right
  • Ca 9 0 ∘ counterclockwise rotation about the point 𝐷
  • Da translation six left
  • Ea 1 8 0 ∘ counterclockwise rotation about the point 𝐷

Hence, are triangles 𝐴 𝐡 𝐢 and 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² similar?

  • Ano
  • Byes
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