Rotations of Shapes - Kuta Software - Free Printable

You’ve uploaded an image showing a geometry worksheet from KutaSoftware: Geometry — All Transformations 2/2, specifically Problem #9:

> 9) rotation 90° clockwise about the origin
> Points:
> - B(–2, 0)
> - C(–4, 3)
> - Z(–3, 4)
> - X(–1, 4)

The task is to rotate each point 90° clockwise about the origin (0,0) and plot the new points.

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## ✔ Step-by-Step Solution

🔁 Rule for 90° Clockwise Rotation About Origin:

To rotate a point (x, y) 90° clockwise about the origin, use the transformation:

> (x, y) → (y, –x)

This is a standard rule in coordinate geometry. Let’s apply it to each point.

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1. Point B(–2, 0)

Apply rule: (x, y) → (y, –x)
→ (0, –(–2)) = (0, 2)

✔ New point: B’(0, 2)

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2. Point C(–4, 3)

Apply rule: (x, y) → (y, –x)
→ (3, –(–4)) = (3, 4)

✔ New point: C’(3, 4)

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3. Point Z(–3, 4)

Apply rule: (x, y) → (y, –x)
→ (4, –(–3)) = (4, 3)

✔ New point: Z’(4, 3)

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4. Point X(–1, 4)

Apply rule: (x, y) → (y, –x)
→ (4, –(–1)) = (4, 1)

✔ New point: X’(4, 1)

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## 📊 Final Answer (Rotated Points):

| Original Point | Rotated 90° Clockwise |
|----------------|------------------------|
| B(–2, 0) | B’(0, 2) |
| C(–4, 3) | C’(3, 4) |
| Z(–3, 4) | Z’(4, 3) |
| X(–1, 4) | X’(4, 1) |

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## 🖍️ How to Plot on the Grid:

On the provided grid (which you can see in the image), locate:

- B’(0, 2) → On the y-axis, 2 units up.
- C’(3, 4) → Right 3, up 4.
- Z’(4, 3) → Right 4, up 3.
- X’(4, 1) → Right 4, up 1.

Connect them in order (B’ → C’ → Z’ → X’) to see the rotated figure.

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## 💡 Why This Rule Works:

Imagine rotating the point (x, y) 90° clockwise:

- The x-coordinate becomes the *new y-coordinate*, but flipped sign (because clockwise rotation moves the point downward relative to its original x).
- The y-coordinate becomes the *new x-coordinate*.

Example: (1, 0) → (0, –1) → that’s correct: rotating (1,0) 90° clockwise lands at (0,–1).

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## ✔ Summary

You’ve successfully rotated all four points 90° clockwise about the origin using the rule (x, y) → (y, –x). The new coordinates are:

> B’(0, 2), C’(3, 4), Z’(4, 3), X’(4, 1)

Plot these on your grid to complete the problem!

Let me know if you want help with Problem #10 (reflection) too! 😊