Rotations exercise from Corbettmaths, including examples and a QR code for video tutorial.

It looks like you've uploaded a worksheet from Corbettmaths on rotations, specifically Video 275 about rotating shapes around a point $ P $. The task is to rotate each shape (a) through (f) as instructed, using point $ P $ as the center of rotation.

Since I can't see the actual image (you mentioned uploading one, but it's not visible in this interface), I'll walk you through how to solve these types of rotation problems step by step. This will help you complete Question 1(a) through (f) accurately.

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🔷 General Method for Rotating Shapes

To rotate a shape around a point $ P $:

1. Identify the coordinates of each vertex of the shape and the center of rotation $ P $.
2. Apply the rotation rules:
- 90° clockwise:
$$
(x, y) \rightarrow (y, -x) \quad \text{relative to } P
$$
- 90° anticlockwise:
$$
(x, y) \rightarrow (-y, x) \quad \text{relative to } P
$$
3. Translate back to absolute grid coordinates by adding the coordinates of $ P $.
4. Plot the new points and draw the rotated shape.

> ✔ Remember: These transformations are applied relative to the center of rotation $ P $. So first, shift everything so that $ P $ is at the origin, apply the rotation, then shift back.

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Let’s go through each part assuming typical positions (based on common Corbettmaths problems). Since no exact coordinates are given, I’ll explain with general steps and give examples.

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🟦 (a) Rotate rectangle A 90° clockwise about P

Assume:
- Rectangle A has vertices at (say) (6,4), (8,4), (8,6), (6,6)
- Point $ P $ is at (4,3)

#### Step-by-step:
1. For each vertex, subtract $ P $'s coordinates:
- (6,4) → (6−4, 4−3) = (2,1)
- (8,4) → (4,1)
- (8,6) → (4,3)
- (6,6) → (2,3)

2. Apply 90° clockwise rotation:
- (x,y) → (y, −x)
- (2,1) → (1, −2)
- (4,1) → (1, −4)
- (4,3) → (3, −4)
- (2,3) → (3, −2)

3. Add back $ P $’s coordinates (4,3):
- (1,−2) + (4,3) = (5,1)
- (1,−4) + (4,3) = (5,−1)
- (3,−4) + (4,3) = (7,−1)
- (3,−2) + (4,3) = (7,1)

4. Plot these new points and connect them to form the rotated rectangle.

✔ Result: The rectangle is now rotated 90° clockwise around $ P $.

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🟦 (b) Rotate triangle B 90° anticlockwise about P

Assume:
- Triangle B has vertices at (6,5), (8,5), (6,7)
- $ P = (4,3) $

#### Steps:
1. Translate relative to $ P $:
- (6,5) → (2,2)
- (8,5) → (4,2)
- (6,7) → (2,4)

2. Apply 90° anticlockwise: (x,y) → (−y,x)
- (2,2) → (−2,2)
- (4,2) → (−2,4)
- (2,4) → (−4,2)

3. Add back $ P $:
- (−2,2)+(4,3) = (2,5)
- (−2,4)+(4,3) = (2,7)
- (−4,2)+(4,3) = (0,5)

4. Plot and draw the new triangle.

✔ New triangle has vertices at (2,5), (2,7), (0,5)

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🟦 (c) Rotate triangle C 90° clockwise about P

Assume:
- Triangle C has vertices at (6,7), (8,7), (7,9)
- $ P = (7,5) $

1. Relative to $ P $:
- (6,7) → (−1,2)
- (8,7) → (1,2)
- (7,9) → (0,4)

2. 90° clockwise: (x,y) → (y, −x)
- (−1,2) → (2,1)
- (1,2) → (2,−1)
- (0,4) → (4,0)

3. Add $ P $: (7,5)
- (2,1)+(7,5) = (9,6)
- (2,−1)+(7,5) = (9,4)
- (4,0)+(7,5) = (11,5)

4. Draw triangle with new vertices.

✔ Rotated triangle now at (9,6), (9,4), (11,5)

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🟦 (d) Rotate shape D 90° clockwise about P

Shape D is an L-shaped figure.

Approach:
- Find all corner points of D.
- Repeat the same process: translate → rotate → translate back.
- Each vertex must be transformed individually.

Example:
Suppose D has vertices: (4,4), (6,4), (6,5), (5,5), (5,6), (4,6)

$ P = (4,3) $

Relative to $ P $:
- (4,4) → (0,1)
- (6,4) → (2,1)
- (6,5) → (2,2)
- (5,5) → (1,2)
- (5,6) → (1,3)
- (4,6) → (0,3)

Rotate 90° clockwise: (x,y) → (y, −x)
- (0,1) → (1,0)
- (2,1) → (1,−2)
- (2,2) → (2,−2)
- (1,2) → (2,−1)
- (1,3) → (3,−1)
- (0,3) → (3,0)

Add $ P = (4,3) $
- (1,0)+(4,3) = (5,3)
- (1,−2)+(4,3) = (5,1)
- (2,−2)+(4,3) = (6,1)
- (2,−1)+(4,3) = (6,2)
- (3,−1)+(4,3) = (7,2)
- (3,0)+(4,3) = (7,3)

Now plot these points and connect them in order.

✔ You get a rotated L-shape.

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🟦 (e) & (f) – Not shown

These likely involve more complex shapes or different rotations. But the method remains the same.

For any shape:
1. Label all vertices.
2. Use vector method: $ \vec{v} = \text{point} - P $
3. Apply rotation matrix:
- 90° clockwise: $ \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} $
- 90° anticlockwise: $ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $
4. Add $ P $ back.

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✔ Summary Tips

| Rotation | Rule |
|--------|------|
| 90° clockwise | $ (x,y) \to (y, -x) $ relative to P |
| 90° anticlockwise | $ (x,y) \to (-y, x) $ relative to P |
| 180° | $ (x,y) \to (-x, -y) $ |
| 270° clockwise / 90° anticlockwise | Same rule |

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🔧 How to Solve on Grid Paper

1. Use tracing paper (or a transparent sheet).
2. Place it over the shape.
3. Mark point $ P $.
4. Rotate the paper 90° in the required direction.
5. Trace the new position of the shape.
6. Remove the paper and draw the rotated image.

This is often the easiest way in exams.

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✔ Final Advice

Since this is a Corbettmaths exercise, they provide templates at the end. After doing each rotation:
- Draw the new shape clearly.
- Label it appropriately (e.g., A', B').
- Ensure all corners are correctly placed.

If you'd like, you can describe the exact coordinates of the shapes and point $ P $, and I can give you the exact answers for each part!

Let me know if you want me to generate a full solution with diagrams (as text descriptions) or help you fill in the blanks.