Worksheet: Rotations on the Coordinate Plane

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 𝐴𝐡𝐢 after a clockwise rotation of 90∘ about the origin.

  • A𝐴′(βˆ’4,8), 𝐡′(βˆ’3,3), 𝐢′(7,βˆ’3)
  • B𝐴′(4,βˆ’8), 𝐡′(βˆ’3,3), 𝐢′(βˆ’7,3)
  • C𝐴′(4,βˆ’8), 𝐡′(3,βˆ’3), 𝐢′(7,βˆ’3)
  • D𝐴′(βˆ’4,8), 𝐡′(3,βˆ’3), 𝐢′(βˆ’7,3)
  • E𝐴′(βˆ’4,8), 𝐡′(βˆ’3,3), 𝐢′(βˆ’7,3)

Q2:

Given that 𝑋(4,βˆ’2), π‘Œ(8,βˆ’3), and 𝑍(5,βˆ’8) form a triangle, determine the images of its vertices after a clock wise rotation of 180∘ about the origin.

  • A𝑋′(2,βˆ’4), π‘Œβ€²(3,βˆ’8), 𝑍′(8,βˆ’5)
  • B𝑋′(βˆ’4,2), π‘Œβ€²(3,βˆ’8), 𝑍′(βˆ’5,8)
  • C𝑋′(2,βˆ’4), π‘Œβ€²(βˆ’8,3), 𝑍′(βˆ’5,8)
  • D𝑋′(βˆ’4,2), π‘Œβ€²(βˆ’8,3), 𝑍′(8,βˆ’5)
  • E𝑋′(βˆ’4,2), π‘Œβ€²(βˆ’8,3), 𝑍′(βˆ’5,8)

Q3:

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

  • A(βˆ’3,βˆ’14)
  • B(βˆ’14,3)
  • C(3,14)
  • D(βˆ’14,βˆ’3)

Q4:

What is the image of 𝐴𝐡𝐢𝐷 under the transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯)?

  • A𝐴′(4,βˆ’3), 𝐡′(5,βˆ’3), 𝐢′(5,βˆ’4), 𝐷′(4,βˆ’5)
  • B𝐴′(βˆ’4,3), 𝐡′(βˆ’5,3), 𝐢′(βˆ’5,4), 𝐷′(βˆ’4,5)
  • C𝐴′(4,5), 𝐡′(5,5), 𝐢′(5,6), 𝐷′(4,7)
  • D𝐴′(4,0), 𝐡′(5,0), 𝐢′(5,1), 𝐷′(4,2)
  • E𝐴′(βˆ’3,4), 𝐡′(βˆ’3,5), 𝐢′(βˆ’4,5), 𝐷′(βˆ’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(βˆ’1,2),(βˆ’2,3), and (βˆ’4,2)
  • B(1,2),(2,3), and (2,βˆ’4)
  • C(βˆ’2,1),(βˆ’2,3), and (4,βˆ’2)
  • D(2,βˆ’1),(3,βˆ’2), and (2,βˆ’4)
  • E(βˆ’1,2),(2,3), and (2,βˆ’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(βˆ’2,1),(βˆ’2,4), and (βˆ’5,2)
  • D(1,βˆ’2),(4,βˆ’2), and (2,βˆ’5)
  • E(2,βˆ’1),(4,βˆ’2), and (2,βˆ’5)

Q7:

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

  • A𝐴′(8,βˆ’7), 𝐡′(3,βˆ’7), 𝐢′(4,βˆ’3)
  • B𝐴′(βˆ’7,8), 𝐡′(βˆ’7,3), 𝐢′(βˆ’3,4)
  • C𝐴′(8,βˆ’7), 𝐡′(βˆ’7,3), 𝐢′(4,βˆ’3)
  • D𝐴′(8,βˆ’7), 𝐡′(3,βˆ’7), 𝐢′(βˆ’3,4)
  • E𝐴′(βˆ’7,8), 𝐡′(3,βˆ’7), 𝐢′(4,βˆ’3)

Q8:

Given that 𝑋(4,βˆ’4), π‘Œ(8,βˆ’5), and 𝑍(5,βˆ’6), form a triangle, determine the images of its vertices after a clockwise rotation of 90∘ about the origin.

  • A𝑋′(βˆ’4,βˆ’4), π‘Œβ€²(βˆ’5,βˆ’8), 𝑍′(6,5)
  • B𝑋′(4,4), π‘Œβ€²(βˆ’5,βˆ’8), 𝑍′(βˆ’6,βˆ’5)
  • C𝑋′(βˆ’4,βˆ’4), π‘Œβ€²(5,8), 𝑍′(βˆ’6,βˆ’5)
  • D𝑋′(βˆ’4,βˆ’4), π‘Œβ€²(βˆ’5,βˆ’8), 𝑍′(βˆ’6,βˆ’5)
  • E𝑋′(4,4), π‘Œβ€²(5,8), 𝑍′(6,5)

Q9:

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

  • A𝐴′(βˆ’3,7), 𝐡′(3,βˆ’4), 𝐢′(βˆ’5,6)
  • B𝐴′(βˆ’3,7), 𝐡′(βˆ’4,3), 𝐢′(6,βˆ’5)
  • C𝐴′(7,βˆ’3), 𝐡′(βˆ’4,3), 𝐢′(βˆ’5,6)
  • D𝐴′(7,βˆ’3), 𝐡′(3,βˆ’4), 𝐢′(6,βˆ’5)
  • E𝐴′(βˆ’3,7), 𝐡′(βˆ’4,3), 𝐢′(βˆ’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)?

  • A90∘ clockwise around the origin
  • B90∘ counterclockwise around the origin
  • C180∘ clockwise or counterclockwise around the origin

Q11:

The point 𝐴(3,0) is rotated about the origin by πœƒ 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 𝐡 have coordinates (βˆ’5,1) and (βˆ’2,1) respectively. 𝐴𝐡 is rotated about the origin by 270∘ counterclockwise to 𝐴′𝐡′.

Determine the coordinates of 𝐴′ and 𝐡′.

  • A𝐴′=(1,βˆ’5), 𝐡′=(1,βˆ’2)
  • B𝐴′=(5,1), 𝐡′=(2,1)
  • C𝐴′=(1,5), 𝐡′=(1,2)
  • D𝐴′=(βˆ’1,βˆ’5), 𝐡′=(βˆ’1,βˆ’2)
  • E𝐴′=(βˆ’1,5), 𝐡′=(βˆ’1,2)

Is the length of 𝐴𝐡 greater than, less than, or equal to the length of 𝐴′𝐡′?

  • AEqual to
  • BGreater than
  • CLess than

Q13:

Describe the single transformation that would map triangle 𝐴𝐡𝐢 onto triangle 𝐴′𝐡′𝐢′ 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,βˆ’2), (5,βˆ’1), (5,βˆ’4)
  • B(βˆ’3,2), (βˆ’5,1), (βˆ’5,4)
  • C(βˆ’3,βˆ’2), (βˆ’5,βˆ’1), (βˆ’5,βˆ’4)
  • D(βˆ’2,βˆ’3), (βˆ’4,βˆ’5), (βˆ’1,βˆ’5)
  • E(3,2), (5,1), (5,4)

Q15:

Rotate triangle 𝐴𝐡𝐢180∘ clockwise about the origin, and state the coordinates of the image.

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

Q16:

Rotate triangle 𝐴𝐡𝐢90∘ clockwise about the origin, and state the coordinates of the image of vertices.

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

Q17:

⃖⃗𝐴𝐡 and ⃖⃗𝐢𝐷 are parallel. Both have been rotated 90∘ clockwise about the point 𝐸 to ⃖⃗𝐴′𝐡′ and ⃖⃗𝐢′𝐷′ respectively. What do you notice about ⃖⃗𝐴′𝐡′ and ⃖⃗𝐢′𝐷′?

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

Q18:

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

  • Ano
  • Byes

Q19:

A rotation about 𝑋 takes 𝑍 to 𝑍′ and π‘Œ to π‘Œβ€². What is the angle of rotation? If 𝑋𝑍=45, what is 𝑋𝑍′?

  • A90∘, 90
  • B53∘, 45
  • C37∘, 45
  • D127∘, 45

Q20:

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

  • Afalse
  • Btrue

Q21:

The vertices of △𝐴𝐡𝐢 are 𝐴(5,βˆ’9), 𝐡(8,βˆ’1), and 𝐢(βˆ’2,βˆ’3). Determine the vertices of △𝐴′𝐡′𝐢′, the image of △𝐴𝐡𝐢 after a rotation through 180∘ about the origin.

  • A𝐴′(9,βˆ’5), 𝐡′(1,βˆ’8), 𝐢′(3,2)
  • B𝐴′(βˆ’5,βˆ’9), 𝐡′(βˆ’8,βˆ’1), 𝐢′(2,βˆ’3)
  • C𝐴′(5,9), 𝐡′(8,1), 𝐢′(βˆ’2,3)
  • D𝐴′(βˆ’5,9), 𝐡′(βˆ’8,1), 𝐢′(2,3)

Q22:

Emma 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 (π‘Ž,𝑏) is the image of the point (π‘₯,𝑦) after a anticlockwise rotation of 90∘ about the origin. Find π‘Ž+𝑦 and 𝑏+π‘₯.

  • Aπ‘Ž+𝑦=2π‘₯, 𝑏+π‘₯=0
  • Bπ‘Ž+𝑦=0, 𝑏+π‘₯=2π‘₯
  • Cπ‘Ž+𝑦=2𝑦, 𝑏+π‘₯=0
  • Dπ‘Ž+𝑦=0, 𝑏+π‘₯=0

Q24:

In the figure, ⃖⃗𝐴𝐡 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 𝐡 have coordinates (βˆ’2,5) and (3,5) respectively.

What is the length of 𝐴𝐡?

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

  • A𝐴′=(5,2), 𝐡′=(5,βˆ’3)
  • B𝐴′=(βˆ’5,βˆ’2), 𝐡′=(βˆ’5,3)
  • C𝐴′=(2,5), 𝐡′=(βˆ’3,5)
  • D𝐴′=(βˆ’5,2), 𝐡′=(βˆ’5,βˆ’3)
  • E𝐴′=(5,βˆ’2), 𝐡′=(5,3)

What is the length of 𝐴′𝐡′?

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

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

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