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Worksheet: Rotations on Coordinate Plane

In this worksheet, we will practice rotating a shape through 90, 180, and 270 degrees about the origin and investigating what happens to the coordinates as a result.

Q1:

Determine the images of the vertices of triangle 𝐴 𝐵 𝐶 after a clockwise rotation of 9 0 ∘ 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:

Determine the images of the vertices of triangle 𝐴 𝐵 𝐶 after a clockwise rotation of 9 0 ∘ about the origin.

  • A 𝐴 ′ ( 4 , − 2 ) , 𝐵 ′ ( − 5 , 8 ) , 𝐶 ′ ( 1 , − 7 )
  • B 𝐴 ′ ( − 4 , 2 ) , 𝐵 ′ ( 5 , − 8 ) , 𝐶 ′ ( 1 , − 7 )
  • C 𝐴 ′ ( 4 , − 2 ) , 𝐵 ′ ( 5 , − 8 ) , 𝐶 ′ ( − 1 , 7 )
  • D 𝐴 ′ ( 4 , − 2 ) , 𝐵 ′ ( 5 , − 8 ) , 𝐶 ′ ( 1 , − 7 )
  • E 𝐴 ′ ( − 4 , 2 ) , 𝐵 ′ ( − 5 , 8 ) , 𝐶 ′ ( − 1 , 7 )

Q3:

Determine the images of the vertices of triangle 𝐴 𝐵 𝐶 after a counterclockwise rotation of 9 0 ∘ about the origin.

  • A 𝐴 ′ ( − 7 , − 8 ) , 𝐵 ′ ( 8 , 3 ) , 𝐶 ′ ( − 5 , − 7 )
  • B 𝐴 ′ ( 7 , 8 ) , 𝐵 ′ ( − 8 , − 3 ) , 𝐶 ′ ( − 5 , − 7 )
  • C 𝐴 ′ ( − 7 , − 8 ) , 𝐵 ′ ( − 8 , − 3 ) , 𝐶 ′ ( 5 , 7 )
  • D 𝐴 ′ ( − 7 , − 8 ) , 𝐵 ′ ( − 8 , − 3 ) , 𝐶 ′ ( − 5 , − 7 )
  • E 𝐴 ′ ( 7 , 8 ) , 𝐵 ′ ( 8 , 3 ) , 𝐶 ′ ( 5 , 7 )

Q4:

Determine the images of the vertices of triangle 𝐴 𝐵 𝐶 after a counterclockwise rotation of 9 0 ∘ about the origin.

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

Q5:

What is the image of 𝐴 𝐵 𝐶 𝐷 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 , 0 ) , 𝐵 ′ ( 5 , 0 ) , 𝐶 ′ ( 5 , 1 ) , 𝐷 ′ ( 4 , 2 )
  • D 𝐴 ′ ( − 3 , 4 ) , 𝐵 ′ ( − 3 , 5 ) , 𝐶 ′ ( − 4 , 5 ) , 𝐷 ′ ( − 5 , 4 )
  • E 𝐴 ′ ( 4 , 5 ) , 𝐵 ′ ( 5 , 5 ) , 𝐶 ′ ( 5 , 6 ) , 𝐷 ′ ( 4 , 7 )

Q6:

A triangle has vertices at the points seen in the figure. Rotate the triangle 9 0 ∘ 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 )

Q7:

Rotate triangle 𝐴 𝐵 𝐶 1 8 0 ∘ 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 ( − 2 , 0 ) , ( 1 , − 2 ) , ( 1 , − 3 )
  • D ( 2 , 0 ) , ( − 1 , − 2 ) , ( − 1 , 3 )
  • E ( 0 , 2 ) , ( − 2 , − 1 ) , ( 3 , − 1 )

Q8:

Two points 𝐴 and 𝐵 have coordinates ( − 5 , 1 ) and ( − 2 , 1 ) respectively. 𝐴 𝐵 is rotated 2 7 0 ∘ 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 𝐴 𝐵 greater than, less than, or equal to the length of 𝐴 ′ 𝐵 ′ ?

  • Aequal to
  • Bless than
  • Cgreater than

Q9:

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

Q10:

A triangle graphed on the coordinate plane has a vertex at ( 3 , 0 ) . Which of the following rotations would move the vertex to point ( − 3 , 0 ) ?

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

Q11:

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

  • A a rotation of 9 0 ∘ clockwise about 𝐸
  • B a rotation of 9 0 ∘ clockwise about 𝐷
  • C a rotation of 2 7 0 ∘ clockwise about 𝐸
  • D a rotation of 9 0 ∘ counterclockwise about 𝐷
  • Ea rotation of 2 7 0 ∘ counterclockwise about 𝐷

Q12:

Rotate triangle 𝐴 𝐵 𝐶 9 0 ∘ clockwise about the origin, and state the coordinates of the image.

  • 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 )

Q13:

⃖ ⃗ 𝐴 𝐵 and ⃖ ⃗ 𝐶 𝐷 are parallel. Both have been rotated 9 0 ∘ clockwise about the point 𝐸 to ⃖ ⃗ 𝐴 ′ 𝐵 ′ and ⃖ ⃗ 𝐶 ′ 𝐷 ′ respectively. What do you notice about ⃖ ⃗ 𝐴 ′ 𝐵 ′ and ⃖ ⃗ 𝐶 ′ 𝐷 ′ ?

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

Q14:

In the given figure, a rotation of 9 0 ∘ counterclockwise about the point 𝐵 ′ would map triangle 𝐴 𝐵 ′ 𝐶 to triangle 𝐴 ′ 𝐵 ′ 𝐶 ′ . Does it follow that the two triangles are congruent?

  • A yes
  • B no

Q15:

A rotation about 𝑋 takes 𝑍 to 𝑍 ′ and 𝑌 to 𝑌 ′ . What is the angle of rotation? If 𝑋 𝑍 = 4 5 , what is 𝑋 𝑍 ′ ?

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

Q16:

Given that 𝑋 ( 4 , − 2 ) , 𝑌 ( 8 , − 3 ) , and 𝑍 ( 5 , − 8 ) form a triangle, determine the images of its vertices after a clockwise rotation of 1 8 0 ∘ about the origin.

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

Q17:

Determine the coordinates of the vertices’ images of triangle 𝐴 𝐵 𝐶 after a counterclockwise rotation of 2 7 0 ∘ around the origin.

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

Q18:

Determine the coordinates of the vertices’ images of triangle 𝐴 𝐵 𝐶 after a counterclockwise rotation of 2 7 0 ∘ around the origin.

  • A 𝐴 ′ ( − 4 , 8 ) , 𝐵 ′ ( 4 , − 2 ) , 𝐶 ′ ( − 8 , 2 )
  • B 𝐴 ′ ( 8 , − 4 ) , 𝐵 ′ ( − 2 , 4 ) , 𝐶 ′ ( − 8 , 2 )
  • C 𝐴 ′ ( − 4 , 8 ) , 𝐵 ′ ( − 2 , 4 ) , 𝐶 ′ ( 2 , − 8 )
  • D 𝐴 ′ ( − 4 , 8 ) , 𝐵 ′ ( − 2 , 4 ) , 𝐶 ′ ( − 8 , 2 )
  • E 𝐴 ′ ( 8 , − 4 ) , 𝐵 ′ ( 4 , − 2 ) , 𝐶 ′ ( 2 , − 8 )

Q19:

Determine the coordinates of the vertices’ images of triangle 𝐴 𝐵 𝐶 after a clockwise rotation of 2 7 0 ∘ around the origin.

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

Q20:

Determine the coordinates of the vertices’ images of triangle 𝐴 𝐵 𝐶 after a clockwise rotation of 2 7 0 ∘ around the origin.

  • A 𝐴 ′ ( 2 , − 8 ) , 𝐵 ′ ( − 3 , 2 ) , 𝐶 ′ ( 6 , − 4 )
  • B 𝐴 ′ ( − 8 , 2 ) , 𝐵 ′ ( 2 , − 3 ) , 𝐶 ′ ( 6 , − 4 )
  • C 𝐴 ′ ( 2 , − 8 ) , 𝐵 ′ ( 2 , − 3 ) , 𝐶 ′ ( − 4 , 6 )
  • D 𝐴 ′ ( 2 , − 8 ) , 𝐵 ′ ( 2 , − 3 ) , 𝐶 ′ ( 6 , − 4 )
  • E 𝐴 ′ ( − 8 , 2 ) , 𝐵 ′ ( − 3 , 2 ) , 𝐶 ′ ( − 4 , 6 )

Q21:

Determine the coordinates of the vertices’ images of triangle 𝐴 𝐵 𝐶 after a counterclockwise rotation of 1 8 0 ∘ around the origin.

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