Rotations of Shapes worksheet with graph paper and triangle figures for practicing geometric transformations.

Let’s solve each rotation problem step by step. We’ll use the rules for rotating points around the origin (0,0) on a coordinate grid.

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Rotation Rules (around origin):

- 90° clockwise:
Original point (x, y) → becomes (y, -x)

- 90° counterclockwise:
Original point (x, y) → becomes (-y, x)

- 180° (either direction):
Original point (x, y) → becomes (-x, -y)

We’ll apply these to each shape’s vertices.

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Problem 1: Rotate 90° clockwise about the origin

Original triangle vertices (from graph):

Looking at the first graph, the triangle has vertices at:

- A = (-4, -2)
- B = (-1, -3)
- C = (-2, 0)

Apply 90° clockwise rule: (x, y) → (y, -x)

- A’ = (-2, -(-4)) = (-2, 4)
- B’ = (-3, -(-1)) = (-3, 1)
- C’ = (0, -(-2)) = (0, 2)

✔ New coordinates: (-2, 4), (-3, 1), (0, 2)

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Problem 2: Rotate 90° counterclockwise about the origin

Original triangle vertices (from second graph):

Looks like:

- D = (2, 1)
- E = (5, 2)
- F = (3, 4)

Apply 90° CCW rule: (x, y) → (-y, x)

- D’ = (-1, 2)
- E’ = (-2, 5)
- F’ = (-4, 3)

✔ New coordinates: (-1, 2), (-2, 5), (-4, 3)

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Problem 3: Rotate 90° clockwise about the origin

Original quadrilateral vertices (third graph):

From the graph:

- G = (-4, 1)
- H = (-1, 1)
- I = (-1, -1)
- J = (-3, -2)

Apply 90° CW: (x, y) → (y, -x)

- G’ = (1, -(-4)) = (1, 4)
- H’ = (1, -(-1)) = (1, 1)
- I’ = (-1, -(-1)) = (-1, 1)
- J’ = (-2, -(-3)) = (-2, 3)

✔ New coordinates: (1, 4), (1, 1), (-1, 1), (-2, 3)

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Problem 4: Rotate 180° about the origin

Original triangle vertices (fourth graph):

Looks like:

- K = (1, 1)
- L = (3, 1)
- M = (2, 3)

Apply 180° rule: (x, y) → (-x, -y)

- K’ = (-1, -1)
- L’ = (-3, -1)
- M’ = (-2, -3)

✔ New coordinates: (-1, -1), (-3, -1), (-2, -3)

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Problem 5: Rotate 90° clockwise about the origin

Given points:

- P(1, -4)
- Q(2, -6)
- R(3, -3)
- S(2, -3)

Apply 90° CW: (x, y) → (y, -x)

- P’ = (-4, -1)
- Q’ = (-6, -2)
- R’ = (-3, -3)
- S’ = (-3, -2)

✔ New coordinates: (-4, -1), (-6, -2), (-3, -3), (-3, -2)

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Problem 6: Rotate 180° about the origin

Given points:

- T(2, 4)
- U(1, 2)
- V(5, 4)

Apply 180° rule: (x, y) → (-x, -y)

- T’ = (-2, -4)
- U’ = (-1, -2)
- V’ = (-5, -4)

✔ New coordinates: (-2, -4), (-1, -2), (-5, -4)

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Final Answer:

1. After 90° clockwise: (-2, 4), (-3, 1), (0, 2)
2. After 90° counterclockwise: (-1, 2), (-2, 5), (-4, 3)
3. After 90° clockwise: (1, 4), (1, 1), (-1, 1), (-2, 3)
4. After 180°: (-1, -1), (-3, -1), (-2, -3)
5. After 90° clockwise: (-4, -1), (-6, -2), (-3, -3), (-3, -2)
6. After 180°: (-2, -4), (-1, -2), (-5, -4)