Math worksheet on geometric rotations, featuring six exercises involving rotation of shapes on a coordinate plane about the origin.

Let’s solve each rotation problem one by one. We’ll use the rules for rotating points around the origin (0,0).

Rotation Rules:

- 90° counterclockwise (ccw): (x, y) → (-y, x)
- 90° clockwise: (x, y) → (y, -x)
- 180°: (x, y) → (-x, -y)

We’ll apply these to each shape’s key points and then sketch or describe the new position.

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Problem 1: Rotation 180° about the origin

Original shape has vertices at:
(-2, 1), (-2, 3), (0, 1) — let’s check from graph.

Actually, looking at the green triangle in #1:
It goes from (-2,1) to (-2,3) to (0,1). So three points.

Apply 180° rule: (x,y) → (-x,-y)

- (-2,1) → (2,-1)
- (-2,3) → (2,-3)
- (0,1) → (0,-1)

So rotated triangle is in quadrant IV, with points (2,-1), (2,-3), (0,-1)

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Problem 2: Rotation 180° about the origin

Green quadrilateral: looks like a trapezoid or irregular quad.

Vertices appear to be:
(1,4), (3,4), (3,2), (1,2)? Wait — no, it's not rectangle.

Looking again:
From graph: top left (1,4), top right (3,4)? No — actually, it seems:

Point A: (1,4)
Point B: (3,4) — but line slopes down? Actually, maybe:

Wait — better to pick clear corners.

Actually, from image:
Shape has points: (1,4), (3,4), (3,2), (1,2) — that would be rectangle, but drawn as slanted? No — wait, in #2, it’s a right trapezoid?

Actually, let me recheck:

In #2: Green shape — starts at (1,4), goes to (3,4), then down to (3,2), then left to (1,2)? But that’s rectangle. But in image, it looks like from (1,4) to (3,4) to (3,2) to (1,2) — yes, rectangle.

But wait — no, in the image, the side from (1,4) to (3,4) is horizontal, then down to (3,2), then left to (1,2), then up to (1,4). Yes, rectangle.

So vertices: (1,4), (3,4), (3,2), (1,2)

Rotate 180°: (x,y) → (-x,-y)

→ (-1,-4), (-3,-4), (-3,-2), (-1,-2)

So rectangle now in third quadrant.

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Problem 3: Rotation 90° ccw about the origin

Green triangle: vertices at (1,5), (1,3), (4,2) — let’s confirm.

From graph: top point (1,5), bottom left (1,3), bottom right (4,2)

Apply 90° ccw: (x,y) → (-y,x)

- (1,5) → (-5,1)
- (1,3) → (-3,1)
- (4,2) → (-2,4)

So new triangle has points (-5,1), (-3,1), (-2,4)

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Problem 4: Rotation 180° about the origin

Green L-shape: Let’s find coordinates.

Looks like: (1,1), (1,3), (3,3), (3,2), (2,2), (2,1) — but simpler to take outer corners.

Actually, it’s a polyomino: from (1,1) to (1,3) to (3,3) to (3,2) to (2,2) to (2,1) back to (1,1)

But for rotation, we can rotate each vertex.

Use 180°: (x,y) → (-x,-y)

So:

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

So rotated shape is mirrored in third quadrant.

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Problem 5: Rotation 90° clockwise about the origin

Green stair-step shape: Let’s list vertices.

Appears to go: (-3,1), (-3,2), (-2,2), (-2,3), (-1,3), (-1,4), (0,4), (0,1) — wait, better to trace.

Actually, from graph: starts at (-3,1), up to (-3,2), right to (-2,2), up to (-2,3), right to (-1,3), up to (-1,4), right to (0,4), down to (0,1), left to (-3,1)? Not quite.

Perhaps key points: (-3,1), (-3,2), (-2,2), (-2,3), (-1,3), (-1,4), (0,4), (0,1) — but (0,1) to (-3,1) is base.

For rotation, we can rotate each corner.

Rule: 90° clockwise: (x,y) → (y, -x)

Take points:

- (-3,1) → (1, 3)
- (-3,2) → (2, 3)
- (-2,2) → (2, 2)
- (-2,3) → (3, 2)
- (-1,3) → (3, 1)
- (-1,4) → (4, 1)
- (0,4) → (4, 0)
- (0,1) → (1, 0)

So new shape will be in first quadrant, starting at (1,3), etc.

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Problem 6: Rotation 90° ccw about the origin

Green zigzag shape: Vertices?

From graph: appears to have points: (1,5), (2,5), (2,4), (3,4), (3,3), (4,3), (4,2), (1,2) — let’s see.

Actually, tracing: start at (1,5), right to (2,5), down to (2,4), right to (3,4), down to (3,3), right to (4,3), down to (4,2), left to (1,2), up to (1,5)? Not exactly — probably closed shape.

Better to take all turning points.

Assume vertices: (1,5), (2,5), (2,4), (3,4), (3,3), (4,3), (4,2), (1,2)

Apply 90° ccw: (x,y) → (-y,x)

- (1,5) → (-5,1)
- (2,5) → (-5,2)
- (2,4) → (-4,2)
- (3,4) → (-4,3)
- (3,3) → (-3,3)
- (4,3) → (-3,4)
- (4,2) → (-2,4)
- (1,2) → (-2,1)

So rotated shape is in second quadrant.

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Now, since the task is to “solve” the problems, and they are graphical rotations, the expected answer is likely the description of where the shape ends up or its new coordinates.

But since this is a worksheet, and user didn’t specify output format, I’ll provide the transformed coordinates for each problem as the solution.

However, note: in many such worksheets, students are expected to draw the rotated figure. But since we’re text-based, we’ll give the new coordinates.

Let me double-check each calculation.

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Rechecking Problem 1:

Points: (-2,1), (-2,3), (0,1)

180°: (-x,-y)

→ (2,-1), (2,-3), (0,-1) ✔

Problem 2:

Points: (1,4), (3,4), (3,2), (1,2)

180°: (-1,-4), (-3,-4), (-3,-2), (-1,-2) ✔

Problem 3:

Points: (1,5), (1,3), (4,2)

90° ccw: (-y,x)

→ (-5,1), (-3,1), (-2,4) ✔

Problem 4:

Points: Let’s simplify — the L-shape can be represented by corners: (1,1), (1,3), (3,3), (3,2), (2,2), (2,1)

180°: (-1,-1), (-1,-3), (-3,-3), (-3,-2), (-2,-2), (-2,-1) ✔

Problem 5:

Points: Let’s take the outer path: (-3,1), (-3,2), (-2,2), (-2,3), (-1,3), (-1,4), (0,4), (0,1)

90° clockwise: (y, -x)

- (-3,1) → (1, 3)
- (-3,2) → (2, 3)
- (-2,2) → (2, 2)
- (-2,3) → (3, 2)
- (-1,3) → (3, 1)
- (-1,4) → (4, 1)
- (0,4) → (4, 0)
- (0,1) → (1, 0)

✔

Problem 6:

Points: (1,5), (2,5), (2,4), (3,4), (3,3), (4,3), (4,2), (1,2)

90° ccw: (-y,x)

- (1,5) → (-5,1)
- (2,5) → (-5,2)
- (2,4) → (-4,2)
- (3,4) → (-4,3)
- (3,3) → (-3,3)
- (4,3) → (-3,4)
- (4,2) → (-2,4)
- (1,2) → (-2,1)

✔

All calculations verified.

Final Answer:
The rotated figures have the following key vertices after transformation:

1) (2,-1), (2,-3), (0,-1)
2) (-1,-4), (-3,-4), (-3,-2), (-1,-2)
3) (-5,1), (-3,1), (-2,4)
4) (-1,-1), (-1,-3), (-3,-3), (-3,-2), (-2,-2), (-2,-1)
5) (1,3), (2,3), (2,2), (3,2), (3,1), (4,1), (4,0), (1,0)
6) (-5,1), (-5,2), (-4,2), (-4,3), (-3,3), (-3,4), (-2,4), (-2,1)