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It looks like you've uploaded an image containing a series of geometric transformation exercises, including rotations, reflections, translations, and some vector-related problems. Let me guide you through solving a few of these problems.

### 1. **Question 6 - Use your rules to solve these transformations**

For example:
- **(a) \( T_{2,-1}(A(5,4)) \)** involves a **translation** by 2 units right and 1 unit down. To compute:
  \[
  A'(5 + 2, 4 - 1) = A'(7, 3)
  \]

- **(d) \( T_{-3, 5}(A(4,-3)) \)**:
  \[
  A'(4 - 3, -3 + 5) = A'(1, 2)
  \]

### 2. **Rotation problem example**

- **\( R_{90^\circ}(A(3, 2)) \)**:
  A 90-degree rotation about the origin involves swapping the coordinates and changing the sign of the new x-coordinate. 
  \[
  A'(x', y') = (-y, x) = (-2, 3)
  \]

### 3. **Reflection Problem**

- **Reflect over the y-axis, \( R_y(A(5,-1)) \)**:
  Reflections over the y-axis reverse the sign of the x-coordinate.
  \[
  A'(5, -1) \rightarrow A'(-5, -1)
  \]

Would you like help with any specific problem from this set?

### Follow-up Questions:
1. How do you determine coordinates after a 180-degree rotation?
2. Can you describe how translation affects the x and y coordinates of a point?
3. How would reflecting over the x-axis change the coordinates of a point?
4. What is the geometric effect of a 270-degree rotation?
5. How can you find the smallest positive angle for a given rotation matrix?

**Tip:** In transformations, always pay attention to the order of operations when combining multiple steps (e.g., reflect and then translate).

Solve geometric transformation problems, including translations, rotations, and reflections on given points.

Solve problems on geometric transformations including translations, rotations, and reflections in coordinate geometry. This exercise covers rules for translating, rotating 90°, 180°, 270°, and reflecting points over axes. Suitable for students in Grades 8-10.

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