Worksheet: Rotations on the Coordinate Plane

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In this worksheet, we will practice finding the vertices of a shape after it undergoes a rotation of 90, 180, or 270 degrees about the origin clockwise and counterclockwise.

Q1:

Determine the images of the vertices of triangle 𝐴𝐵𝐶 after a clockwise rotation of 90∘ about the origin.

  • A 𝐴 ′ ( − 4 , 8 ) , 𝐵 ′ ( − 3 , 3 ) , 𝐶 ′ ( 7 , − 3 )
  • B 𝐴 ′ ( 4 , − 8 ) , 𝐵 ′ ( − 3 , 3 ) , 𝐶 ′ ( − 7 , 3 )
  • C 𝐴 ′ ( 4 , − 8 ) , 𝐵 ′ ( 3 , − 3 ) , 𝐶 ′ ( 7 , − 3 )
  • D 𝐴 ′ ( − 4 , 8 ) , 𝐵 ′ ( 3 , − 3 ) , 𝐶 ′ ( − 7 , 3 )
  • E 𝐴 ′ ( − 4 , 8 ) , 𝐵 ′ ( − 3 , 3 ) , 𝐶 ′ ( − 7 , 3 )

Q2:

Given that 𝑋(4,−2), 𝑌(8,−3), and 𝑍(5,−8) form a triangle, determine the images of its vertices after a clock wise rotation of 180∘ about the origin.

  • A 𝑋 ′ ( 2 , − 4 ) , 𝑌 ′ ( 3 , − 8 ) , 𝑍 ′ ( 8 , − 5 )
  • B 𝑋 ′ ( − 4 , 2 ) , 𝑌 ′ ( 3 , − 8 ) , 𝑍 ′ ( − 5 , 8 )
  • C 𝑋 ′ ( 2 , − 4 ) , 𝑌 ′ ( − 8 , 3 ) , 𝑍 ′ ( − 5 , 8 )
  • D 𝑋 ′ ( − 4 , 2 ) , 𝑌 ′ ( − 8 , 3 ) , 𝑍 ′ ( 8 , − 5 )
  • E 𝑋 ′ ( − 4 , 2 ) , 𝑌 ′ ( − 8 , 3 ) , 𝑍 ′ ( − 5 , 8 )

Q3:

What is the image of the point (−3,14) after a rotation about the origin through an angle of 90∘?

  • A ( − 3 , − 1 4 )
  • B ( − 1 4 , 3 )
  • C ( 3 , 1 4 )
  • D ( − 1 4 , − 3 )

Q4:

What is the image of 𝐴𝐵𝐶𝐷 under the transformation (𝑥,𝑦)→(−𝑦,𝑥)?

  • A 𝐴 ′ ( 4 , − 3 ) , 𝐵 ′ ( 5 , − 3 ) , 𝐶 ′ ( 5 , − 4 ) , 𝐷 ′ ( 4 , − 5 )
  • B 𝐴 ′ ( − 4 , 3 ) , 𝐵 ′ ( − 5 , 3 ) , 𝐶 ′ ( − 5 , 4 ) , 𝐷 ′ ( − 4 , 5 )
  • C 𝐴 ′ ( 4 , 5 ) , 𝐵 ′ ( 5 , 5 ) , 𝐶 ′ ( 5 , 6 ) , 𝐷 ′ ( 4 , 7 )
  • D 𝐴 ′ ( 4 , 0 ) , 𝐵 ′ ( 5 , 0 ) , 𝐶 ′ ( 5 , 1 ) , 𝐷 ′ ( 4 , 2 )
  • E 𝐴 ′ ( − 3 , 4 ) , 𝐵 ′ ( − 3 , 5 ) , 𝐶 ′ ( − 4 , 5 ) , 𝐷 ′ ( − 5 , 4 )

Q5:

A triangle has its vertices at the points (2, 1), (3, 2) and (2, 4). The triangle is rotated 90∘ counterclockwise about the origin. At which of the following coordinates will the image have its vertices?

  • A ( − 1 , 2 ) , ( − 2 , 3 ) , and (−4,2)
  • B ( 1 , 2 ) , ( 2 , 3 ) , and (2,−4)
  • C ( − 2 , 1 ) , ( − 2 , 3 ) , and (4,−2)
  • D ( 2 , − 1 ) , ( 3 , − 2 ) , and (2,−4)
  • E ( − 1 , 2 ) , ( 2 , 3 ) , and (2,−4)

Q6:

Rotate the given triangle about the origin 90∘ clockwise. Which of the following sets of coordinates will be the vertices of the image?

  • A ( 2 , 1 ) , ( 4 , 2 ) , and (2,5)
  • B ( 1 , 2 ) , ( 2 , 4 ) , and (5,2)
  • C ( − 2 , 1 ) , ( − 2 , 4 ) , and (−5,2)
  • D ( 1 , − 2 ) , ( 4 , − 2 ) , and (2,−5)
  • E ( 2 , − 1 ) , ( 4 , − 2 ) , and (2,−5)

Q7:

Determine the coordinates of the vertices’ images of triangle 𝐴𝐵𝐶 after a counterclockwise rotation of 180∘ around the origin.

  • A 𝐴 ′ ( 8 , − 7 ) , 𝐵 ′ ( 3 , − 7 ) , 𝐶 ′ ( 4 , − 3 )
  • B 𝐴 ′ ( − 7 , 8 ) , 𝐵 ′ ( − 7 , 3 ) , 𝐶 ′ ( − 3 , 4 )
  • C 𝐴 ′ ( 8 , − 7 ) , 𝐵 ′ ( − 7 , 3 ) , 𝐶 ′ ( 4 , − 3 )
  • D 𝐴 ′ ( 8 , − 7 ) , 𝐵 ′ ( 3 , − 7 ) , 𝐶 ′ ( − 3 , 4 )
  • E 𝐴 ′ ( − 7 , 8 ) , 𝐵 ′ ( 3 , − 7 ) , 𝐶 ′ ( 4 , − 3 )

Q8:

Given that 𝑋(4,−4), 𝑌(8,−5), and 𝑍(5,−6), form a triangle, determine the images of its vertices after a clockwise rotation of 90∘ about the origin.

  • A 𝑋 ′ ( − 4 , − 4 ) , 𝑌 ′ ( − 5 , − 8 ) , 𝑍 ′ ( 6 , 5 )
  • B 𝑋 ′ ( 4 , 4 ) , 𝑌 ′ ( − 5 , − 8 ) , 𝑍 ′ ( − 6 , − 5 )
  • C 𝑋 ′ ( − 4 , − 4 ) , 𝑌 ′ ( 5 , 8 ) , 𝑍 ′ ( − 6 , − 5 )
  • D 𝑋 ′ ( − 4 , − 4 ) , 𝑌 ′ ( − 5 , − 8 ) , 𝑍 ′ ( − 6 , − 5 )
  • E 𝑋 ′ ( 4 , 4 ) , 𝑌 ′ ( 5 , 8 ) , 𝑍 ′ ( 6 , 5 )

Q9:

Determine the coordinates of the vertices’ images of triangle 𝐴𝐵𝐶 after a counterclockwise rotation of 270∘ around the origin.

  • A 𝐴 ′ ( − 3 , 7 ) , 𝐵 ′ ( 3 , − 4 ) , 𝐶 ′ ( − 5 , 6 )
  • B 𝐴 ′ ( − 3 , 7 ) , 𝐵 ′ ( − 4 , 3 ) , 𝐶 ′ ( 6 , − 5 )
  • C 𝐴 ′ ( 7 , − 3 ) , 𝐵 ′ ( − 4 , 3 ) , 𝐶 ′ ( − 5 , 6 )
  • D 𝐴 ′ ( 7 , − 3 ) , 𝐵 ′ ( 3 , − 4 ) , 𝐶 ′ ( 6 , − 5 )
  • E 𝐴 ′ ( − 3 , 7 ) , 𝐵 ′ ( − 4 , 3 ) , 𝐶 ′ ( − 5 , 6 )

Q10:

A triangle graphed on the coordinate plane has a vertex at (6,0). Which of the following rotations would move the vertex to point (0,6)?

  • A 9 0 ∘ clockwise around the origin
  • B 9 0 ∘ counterclockwise around the origin
  • C 1 8 0 ∘ clockwise or counterclockwise around the origin

Q11:

The point 𝐴(3,0) is rotated about the origin by 𝜃 degrees. On which of the given figures will the image of 𝐴 lie?

  • Aa circle with radius three and center (0,0)
  • Ba circle with radius three and center (1,1)
  • Ca circle with radius three and center (3,3)
  • Da circle with radius three and center (3,0)
  • Ea circle with radius two and center (0,0)

Q12:

Two points 𝐴 and 𝐵 have coordinates (−5,1) and (−2,1) respectively. 𝐴𝐵 is rotated 270∘ counterclockwise to 𝐴′𝐵′.

Determine the coordinates of 𝐴′ and 𝐵′.

  • A 𝐴 ′ = ( 1 , − 5 ) , 𝐵 ′ = ( 1 , − 2 )
  • B 𝐴 ′ = ( 5 , 1 ) , 𝐵 ′ = ( 2 , 1 )
  • C 𝐴 ′ = ( 1 , 5 ) , 𝐵 ′ = ( 1 , 2 )
  • D 𝐴 ′ = ( − 1 , − 5 ) , 𝐵 ′ = ( − 1 , − 2 )
  • E 𝐴 ′ = ( − 1 , 5 ) , 𝐵 ′ = ( − 1 , 2 )

Is the length of 𝐴𝐵 greater than, less than, or equal to the length of 𝐴′𝐵′?

  • Aequal to
  • Bgreater than
  • Cless than

Q13:

Describe the single transformation that would map triangle 𝐴𝐵𝐶 onto triangle 𝐴′𝐵′𝐶′ in the given figure.

  • Aa rotation of 270∘ counterclockwise about 𝐷
  • Ba rotation of 270∘ clockwise about 𝐸
  • Ca rotation of 90∘ clockwise about 𝐸
  • Da rotation of 90∘ clockwise about 𝐷
  • Ea rotation of 90∘ counterclockwise about 𝐷

Q14:

A triangle has vertices at the points seen in the figure. Rotate the triangle 90∘ counterclockwise about the origin, and determine the coordinates of the image.

  • A ( 3 , − 2 ) , ( 5 , − 1 ) , ( 5 , − 4 )
  • B ( − 3 , 2 ) , ( − 5 , 1 ) , ( − 5 , 4 )
  • C ( − 3 , − 2 ) , ( − 5 , − 1 ) , ( − 5 , − 4 )
  • D ( − 2 , − 3 ) , ( − 4 , − 5 ) , ( − 1 , − 5 )
  • E ( 3 , 2 ) , ( 5 , 1 ) , ( 5 , 4 )

Q15:

Rotate triangle 𝐴𝐵𝐶180∘ clockwise about the origin, and state the coordinates of the image.

  • A ( 2 , 0 ) , ( 1 , − 2 ) , ( 1 , 3 )
  • B ( − 2 , 0 ) , ( 1 , − 2 ) , ( 1 , − 3 )
  • C ( 0 , 2 ) , ( − 2 , − 1 ) , ( 3 , − 1 )
  • D ( − 2 , 0 ) , ( 1 , 2 ) , ( 1 , − 3 )
  • E ( 2 , 0 ) , ( − 1 , − 2 ) , ( − 1 , 3 )

Q16:

Rotate triangle 𝐴𝐵𝐶90∘ clockwise about the origin, and state the coordinates of the image of vertices.

  • A ( 2 , − 5 ) , ( 4 , − 3 ) , ( 0 , − 2 )
  • B ( − 5 , 2 ) , ( − 3 , 4 ) , ( − 2 , 0 )
  • C ( 2 , 5 ) , ( 4 , 3 ) , ( 0 , 2 )
  • D ( − 2 , 5 ) , ( − 4 , 3 ) , ( 0 , 2 )
  • E ( − 2 , − 5 ) , ( − 4 , − 3 ) , ( − 2 , 0 )

Q17:

⃖ ⃗ 𝐴 𝐵 and ⃖⃗𝐶𝐷 are parallel. Both have been rotated 90∘ clockwise about the point 𝐸 to ⃖⃗𝐴′𝐵′ and ⃖⃗𝐶′𝐷′ respectively. What do you notice about ⃖⃗𝐴′𝐵′ and ⃖⃗𝐶′𝐷′?

  • AThey are parallel.
  • BThey are intersecting.
  • CThey are perpendicular.

Q18:

In the given figure, a rotation of 90∘ counterclockwise about the point 𝐵′ would map triangle 𝐴𝐵′𝐶 to triangle 𝐴′𝐵′𝐶′. Does it follow that the two triangles are congruent?

  • Ano
  • Byes

Q19:

A rotation about 𝑋 takes 𝑍 to 𝑍′ and 𝑌 to 𝑌′. What is the angle of rotation? If 𝑋𝑍=45, what is 𝑋𝑍′?

  • A 9 0 ∘ , 90
  • B 5 3 ∘ , 45
  • C 3 7 ∘ , 45
  • D 1 2 7 ∘ , 45

Q20:

In the given figure, if 𝑀 is the midpoint of 𝐴𝐵, then 𝑏 can be rotated 180∘ about 𝑀 to 𝑎. Hence, 𝑎 and 𝑏 must be congruent. Is this statement true or false?

  • Afalse
  • Btrue

Q21:

The vertices of △𝐴𝐵𝐶 are 𝐴(5,−9), 𝐵(8,−1), and 𝐶(−2,−3). Determine the vertices of △𝐴′𝐵′𝐶′, the image of △𝐴𝐵𝐶 after a rotation through 180∘ about the origin.

  • A 𝐴 ′ ( 9 , − 5 ) , 𝐵 ′ ( 1 , − 8 ) , 𝐶 ′ ( 3 , 2 )
  • B 𝐴 ′ ( − 5 , − 9 ) , 𝐵 ′ ( − 8 , − 1 ) , 𝐶 ′ ( 2 , − 3 )
  • C 𝐴 ′ ( 5 , 9 ) , 𝐵 ′ ( 8 , 1 ) , 𝐶 ′ ( − 2 , 3 )
  • D 𝐴 ′ ( − 5 , 9 ) , 𝐵 ′ ( − 8 , 1 ) , 𝐶 ′ ( 2 , 3 )

Q22:

Isabella went on a hike in the woods. She was walking southwest along a trail. When she got to a fork in the trail, she started walking southeast. Fully describe the rotation she made at the fork in the trail.

  • ARotation by 90∘ counterclockwise
  • BRotation by 45∘ counterclockwise
  • CRotation by 180∘ clockwise
  • DRotation by 90∘ clockwise
  • ERotation by 45∘ clockwise

Q23:

The point (𝑎,𝑏) is the image of the point (𝑥,𝑦) after a anticlockwise rotation of 90∘ about the origin. Find 𝑎+𝑦 and 𝑏+𝑥.

  • A 𝑎 + 𝑦 = 2 𝑥 , 𝑏 + 𝑥 = 0
  • B 𝑎 + 𝑦 = 0 , 𝑏 + 𝑥 = 2 𝑥
  • C 𝑎 + 𝑦 = 2 𝑦 , 𝑏 + 𝑥 = 0
  • D 𝑎 + 𝑦 = 0 , 𝑏 + 𝑥 = 0

Q24:

In the figure, ⃖⃗𝐴𝐵 has been rotated about the origin by 180∘. Is the image a point, a line segment, or a line?

  • Aa point
  • Ba line segment
  • Ca line

Q25:

Points 𝐴 and 𝐵 have coordinates (−2,5) and (3,5) respectively.

What is the length of 𝐴𝐵?

A rotation of 90∘ clockwise about the origin maps the points 𝐴 and 𝐵 to the points 𝐴′ and 𝐵′ respectively. What are the coordinates of 𝐴′ and 𝐵′?

  • A 𝐴 ′ = ( 5 , 2 ) , 𝐵 ′ = ( 5 , − 3 )
  • B 𝐴 ′ = ( − 5 , − 2 ) , 𝐵 ′ = ( − 5 , 3 )
  • C 𝐴 ′ = ( 2 , 5 ) , 𝐵 ′ = ( − 3 , 5 )
  • D 𝐴 ′ = ( − 5 , 2 ) , 𝐵 ′ = ( − 5 , − 3 )
  • E 𝐴 ′ = ( 5 , − 2 ) , 𝐵 ′ = ( 5 , 3 )

What is the length of 𝐴′𝐵′?

Hence, does the length of 𝐴𝐵 decrease, increase, or stay the same as a result of this rotation?

  • AIt decreases.
  • BIt stays the same.
  • CIt increases.

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