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 𝐡 𝐢 after a clockwise rotation of 9 0 ∘ about the origin.

  • A 𝐴 β€² ( βˆ’ 4 , 8 ) , 𝐡 β€² ( 3 , βˆ’ 3 ) , 𝐢 β€² ( βˆ’ 7 , 3 )
  • B 𝐴 β€² ( 4 , βˆ’ 8 ) , 𝐡 β€² ( βˆ’ 3 , 3 ) , 𝐢 β€² ( βˆ’ 7 , 3 )
  • C 𝐴 β€² ( βˆ’ 4 , 8 ) , 𝐡 β€² ( βˆ’ 3 , 3 ) , 𝐢 β€² ( 7 , βˆ’ 3 )
  • D 𝐴 β€² ( βˆ’ 4 , 8 ) , 𝐡 β€² ( βˆ’ 3 , 3 ) , 𝐢 β€² ( βˆ’ 7 , 3 )
  • E 𝐴 β€² ( 4 , βˆ’ 8 ) , 𝐡 β€² ( 3 , βˆ’ 3 ) , 𝐢 β€² ( 7 , βˆ’ 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 clockwise rotation of 1 8 0 ∘ about the origin.

  • A 𝑋 β€² ( βˆ’ 4 , 2 ) , π‘Œ β€² ( 3 , βˆ’ 8 ) , 𝑍 β€² ( βˆ’ 5 , 8 )
  • B 𝑋 β€² ( 2 , βˆ’ 4 ) , π‘Œ β€² ( βˆ’ 8 , 3 ) , 𝑍 β€² ( βˆ’ 5 , 8 )
  • C 𝑋 β€² ( βˆ’ 4 , 2 ) , π‘Œ β€² ( βˆ’ 8 , 3 ) , 𝑍 β€² ( 8 , βˆ’ 5 )
  • D 𝑋 β€² ( βˆ’ 4 , 2 ) , π‘Œ β€² ( βˆ’ 8 , 3 ) , 𝑍 β€² ( βˆ’ 5 , 8 )
  • E 𝑋 β€² ( 2 , βˆ’ 4 ) , π‘Œ β€² ( 3 , βˆ’ 8 ) , 𝑍 β€² ( 8 , βˆ’ 5 )

Q3:

What is the image of the point ( βˆ’ 3 , 1 4 ) after a rotation about the origin through an angle of 9 0 ∘ ?

  • A ( 3 , 1 4 )
  • B ( βˆ’ 1 4 , 3 )
  • C ( βˆ’ 3 , βˆ’ 1 4 )
  • D ( βˆ’ 1 4 , βˆ’ 3 )

Q4:

What type of transformation is the following?

  • Atranslation
  • Breflection
  • Crotation

Q5:

What is the image of 𝐴 𝐡 𝐢 𝐷 under the transformation ( π‘₯ , 𝑦 ) β†’ ( βˆ’ 𝑦 , π‘₯ ) ?

  • A 𝐴 β€² ( βˆ’ 4 , 3 ) , 𝐡 β€² ( βˆ’ 5 , 3 ) , 𝐢 β€² ( βˆ’ 5 , 4 ) , 𝐷 β€² ( βˆ’ 4 , 5 )
  • B 𝐴 β€² ( 4 , βˆ’ 3 ) , 𝐡 β€² ( 5 , βˆ’ 3 ) , 𝐢 β€² ( 5 , βˆ’ 4 ) , 𝐷 β€² ( 4 , βˆ’ 5 )
  • C 𝐴 β€² ( 4 , 0 ) , 𝐡 β€² ( 5 , 0 ) , 𝐢 β€² ( 5 , 1 ) , 𝐷 β€² ( 4 , 2 )
  • D 𝐴 β€² ( βˆ’ 3 , 4 ) , 𝐡 β€² ( βˆ’ 3 , 5 ) , 𝐢 β€² ( βˆ’ 4 , 5 ) , 𝐷 β€² ( βˆ’ 5 , 4 )
  • E 𝐴 β€² ( 4 , 5 ) , 𝐡 β€² ( 5 , 5 ) , 𝐢 β€² ( 5 , 6 ) , 𝐷 β€² ( 4 , 7 )

Q6:

Which transformation can turn the letter d into p?

  • Atranslation
  • Breflection
  • Crotation

Q7:

Which transformation can turn the letter Z into N?

  • Atranslation
  • Breflection
  • Crotation

Q8:

Which of the following represents a rotation of the shaded figure?

  • A
  • B
  • C
  • D

Q9:

A triangle has its vertices at the points (2, 1), (3, 2) and (2, 4). The triangle is rotated 9 0 ∘ counterclockwise about the origin. At which of the following coordinates will the image have its vertices?

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

Q10:

Rotate the given triangle about the origin 9 0 ∘ clockwise. Which of the following sets of coordinates will be the vertices of the image?

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

Q11:

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

  • A 𝐴 β€² ( 8 , βˆ’ 7 ) , 𝐡 β€² ( βˆ’ 7 , 3 ) , 𝐢 β€² ( 4 , βˆ’ 3 )
  • B 𝐴 β€² ( βˆ’ 7 , 8 ) , 𝐡 β€² ( 3 , βˆ’ 7 ) , 𝐢 β€² ( 4 , βˆ’ 3 )
  • C 𝐴 β€² ( 8 , βˆ’ 7 ) , 𝐡 β€² ( 3 , βˆ’ 7 ) , 𝐢 β€² ( βˆ’ 3 , 4 )
  • D 𝐴 β€² ( 8 , βˆ’ 7 ) , 𝐡 β€² ( 3 , βˆ’ 7 ) , 𝐢 β€² ( 4 , βˆ’ 3 )
  • E 𝐴 β€² ( βˆ’ 7 , 8 ) , 𝐡 β€² ( βˆ’ 7 , 3 ) , 𝐢 β€² ( βˆ’ 3 , 4 )

Q12:

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 9 0 ∘ about the origin.

  • A 𝑋 β€² ( βˆ’ 4 , βˆ’ 4 ) , π‘Œ β€² ( 5 , 8 ) , 𝑍 β€² ( βˆ’ 6 , βˆ’ 5 )
  • B 𝑋 β€² ( 4 , 4 ) , π‘Œ β€² ( βˆ’ 5 , βˆ’ 8 ) , 𝑍 β€² ( βˆ’ 6 , βˆ’ 5 )
  • C 𝑋 β€² ( βˆ’ 4 , βˆ’ 4 ) , π‘Œ β€² ( βˆ’ 5 , βˆ’ 8 ) , 𝑍 β€² ( 6 , 5 )
  • D 𝑋 β€² ( βˆ’ 4 , βˆ’ 4 ) , π‘Œ β€² ( βˆ’ 5 , βˆ’ 8 ) , 𝑍 β€² ( βˆ’ 6 , βˆ’ 5 )
  • E 𝑋 β€² ( 4 , 4 ) , π‘Œ β€² ( 5 , 8 ) , 𝑍 β€² ( 6 , 5 )

Q13:

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

  • A 𝐴 β€² ( βˆ’ 3 , 7 ) , 𝐡 β€² ( 3 , βˆ’ 4 ) , 𝐢 β€² ( βˆ’ 5 , 6 )
  • B 𝐴 β€² ( 7 , βˆ’ 3 ) , 𝐡 β€² ( βˆ’ 4 , 3 ) , 𝐢 β€² ( βˆ’ 5 , 6 )
  • C 𝐴 β€² ( βˆ’ 3 , 7 ) , 𝐡 β€² ( βˆ’ 4 , 3 ) , 𝐢 β€² ( 6 , βˆ’ 5 )
  • D 𝐴 β€² ( βˆ’ 3 , 7 ) , 𝐡 β€² ( βˆ’ 4 , 3 ) , 𝐢 β€² ( βˆ’ 5 , 6 )
  • E 𝐴 β€² ( 7 , βˆ’ 3 ) , 𝐡 β€² ( 3 , βˆ’ 4 ) , 𝐢 β€² ( 6 , βˆ’ 5 )

Q14:

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

Q15:

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

Q16:

Two points 𝐴 and 𝐡 have coordinates ( βˆ’ 5 , 1 ) and ( βˆ’ 2 , 1 ) respectively. 𝐴 𝐡 is rotated 2 7 0 ∘ counterclockwise to 𝐴 β€² 𝐡 β€² .

Determine the coordinates of 𝐴 β€² and 𝐡 β€² .

  • A 𝐴 β€² = ( 1 , βˆ’ 5 ) , 𝐡 β€² = ( 1 , βˆ’ 2 )
  • B 𝐴 β€² = ( 5 , 1 ) , 𝐡 β€² = ( 2 , 1 )
  • C 𝐴 β€² = ( βˆ’ 1 , 5 ) , 𝐡 β€² = ( βˆ’ 1 , 2 )
  • D 𝐴 β€² = ( 1 , 5 ) , 𝐡 β€² = ( 1 , 2 )
  • E 𝐴 β€² = ( βˆ’ 1 , βˆ’ 5 ) , 𝐡 β€² = ( βˆ’ 1 , βˆ’ 2 )

Is the length of 𝐴 𝐡 greater than, less than, or equal to the length of 𝐴 β€² 𝐡 β€² ?

  • Aequal to
  • Bless than
  • Cgreater than

Q17:

Describe the single transformation that would map triangle 𝐴 𝐡 𝐢 onto triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² 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

Q18:

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 ( βˆ’ 3 , βˆ’ 2 ) , ( βˆ’ 5 , βˆ’ 1 ) , ( βˆ’ 5 , βˆ’ 4 )
  • B ( βˆ’ 3 , 2 ) , ( βˆ’ 5 , 1 ) , ( βˆ’ 5 , 4 )
  • C ( 3 , βˆ’ 2 ) , ( 5 , βˆ’ 1 ) , ( 5 , βˆ’ 4 )
  • D ( βˆ’ 2 , βˆ’ 3 ) , ( βˆ’ 4 , βˆ’ 5 ) , ( βˆ’ 1 , βˆ’ 5 )
  • E ( 3 , 2 ) , ( 5 , 1 ) , ( 5 , 4 )

Q19:

Rotate triangle 𝐴 𝐡 𝐢 1 8 0 ∘ clockwise about the origin, and state the coordinates of the image.

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

Q20:

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

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

Q21:

βƒ–     βƒ— 𝐴 𝐡 and βƒ–     βƒ— 𝐢 𝐷 are parallel. Both have been rotated 9 0 ∘ clockwise about the point 𝐸 to βƒ–        βƒ— 𝐴 β€² 𝐡 β€² and βƒ–         βƒ— 𝐢 β€² 𝐷 β€² respectively. What do you notice about βƒ–        βƒ— 𝐴 β€² 𝐡 β€² and βƒ–         βƒ— 𝐢 β€² 𝐷 β€² ?

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

Q22:

In the given figure, a rotation of 9 0 ∘ counterclockwise about the point 𝐡 β€² would map triangle 𝐴 𝐡 β€² 𝐢 to triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² . Does it follow that the two triangles are congruent?

  • A yes
  • B no

Q23:

A rotation about 𝑋 takes 𝑍 to 𝑍 β€² and π‘Œ to π‘Œ β€² . What is the angle of rotation? If 𝑋 𝑍 = 4 5 , what is 𝑋 𝑍 β€² ?

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

Q24:

In the given figure, if 𝑀 is the midpoint of 𝐴 𝐡 , then 𝑏 can be rotated 1 8 0 ∘ about 𝑀 to π‘Ž . Hence, π‘Ž and 𝑏 must be congruent. Is this statement true or false?

  • A true
  • B false

Q25:

𝐴 𝐡 is rotated 9 0 ∘ clockwise about the origin. Is the length of the image resulting from this transformation greater than, less than, or the same as the length of 𝐴 𝐡 ?

  • Aless than
  • Bgreater than
  • Cthe same

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