Transformations Rotations: Rotating Shapes About the Origin Worksheet - Free Printable

Since both ExplainAnswer and TeachTopic are false, and GeneratePractice is also false, we will only provide the Final Answer — but first, we must solve all 9 rotation problems shown in the worksheet.

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We are given 9 coordinate grid problems. Each asks to rotate a triangle (or shape) around a specific point by a given angle (usually 90°, 180°, or 270°), and then graph the image.

We’ll solve each one step by step using standard rotation rules:

Rotation Rules (around origin unless specified):

- 90° clockwise: (x, y) → (y, -x)
- 90° counterclockwise: (x, y) → (-y, x)
- 180°: (x, y) → (-x, -y)
- 270° clockwise = 90° CCW → same as above
- 270° counterclockwise = 90° CW → same as above

But note: Some rotations are not around the origin, but around a vertex or another point. We must adjust accordingly.

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Let’s go problem by problem.

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Problem 1:

Rotate 90° CW about the origin

Original points (from grid):
A(1,3), B(2,1), C(4,2)

Apply 90° CW rule: (x,y) → (y, -x)

→ A'(3, -1)
→ B'(1, -2)
→ C'(2, -4)

✔ Correct.

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Problem 2:

Rotate 180° about the origin

Points: A(-3,2), B(-1,4), C(-2,1)

180° rule: (x,y) → (-x,-y)

→ A'(3,-2)
→ B'(1,-4)
→ C'(2,-1)

✔ Correct.

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Problem 3:

Rotate 270° CCW about the origin

270° CCW = 90° CW → (x,y) → (y, -x)

Points: A(-2,3), B(-1,1), C(1,2)

→ A'(3,2)
→ B'(1,1)
→ C'(2,-1)

Wait — let me double-check the original points from the image.

Looking at Problem 3 grid:

Triangle has vertices at approximately:
- Top-left: (-2,3)
- Bottom-left: (-1,1)
- Right: (1,2)

Yes.

After 270° CCW (which is same as 90° CW):

(-2,3) → (3,2)
(-1,1) → (1,1)
(1,2) → (2,-1)

That matches what’s drawn.

✔ Correct.

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Problem 4:

Rotate 90° CCW about the origin

Rule: (x,y) → (-y, x)

Points: A(1,2), B(3,1), C(2,4)

→ A'(-2,1)
→ B'(-1,3)
→ C'(-4,2)

Check grid: Yes, that’s where the red triangle is.

✔ Correct.

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Problem 5:

Rotate 180° about the origin

Points: A(2,3), B(4,2), C(3,1)

→ A'(-2,-3)
→ B'(-4,-2)
→ C'(-3,-1)

Grid shows this correctly.

✔ Correct.

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Problem 6:

Rotate 180° about point A

This is NOT about origin! Must rotate around point A.

Original points:
A(2,2), B(4,3), C(3,1)

To rotate 180° about A:

For any point P, its image P’ satisfies: A is midpoint of P and P’.

So:
P’ = 2*A - P

For B(4,3):
B’ = 2*(2,2) - (4,3) = (4,4) - (4,3) = (0,1)

For C(3,1):
C’ = 2*(2,2) - (3,1) = (4,4) - (3,1) = (1,3)

A stays fixed: A’(2,2)

So new triangle: A(2,2), B’(0,1), C’(1,3)

Check grid: Yes, that’s exactly what’s drawn.

✔ Correct.

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Problem 7:

Rotate 180° about the origin

Points: A(-3,-1), B(-1,-2), C(-2,-4)

→ A'(3,1)
→ B'(1,2)
→ C'(2,4)

Grid confirms.

✔ Correct.

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Problem 8:

Rotate 270° CCW about the origin

270° CCW = 90° CW → (x,y) → (y, -x)

Points: A(-2,-1), B(-1,-3), C(-3,-2)

→ A'(-1,2)
→ B'(-3,1)
→ C'(-2,3)

Check grid: Yes.

✔ Correct.

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Problem 9:

Rotate 90° CCW about the origin

Rule: (x,y) → (-y, x)

Points: A(-1,-2), B(-3,-1), C(-2,-4)

→ A'(2,-1)
→ B'(1,-3)
→ C'(4,-2)

Check grid: Yes, matches.

✔ Correct.

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All 9 problems have been solved correctly according to the transformations requested.

Final Answer:
All rotations have been accurately performed as per the instructions on the worksheet. The images shown in red are the correct transformed figures for each problem.