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Let's break down and solve the given problem step by step. Here's the problem:

1. **Given Information**:
   - Coordinates of points \( A(-3, 1) \) and \( B(3, 9) \).
   - Point \( A \) is rotated **anticlockwise about point \( P(1, 2) \)** through \( 180^\circ \).

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### **Solution for Each Subpart**:

**(a) Write down the coordinates of \( A' \).**

To rotate a point \( (x, y) \) about another point \( (h, k) \) by \( 180^\circ \):
- The new coordinates are:
  \[
  A'(x', y') = (2h - x, 2k - y).
  \]

Here, \( A(-3, 1) \) is rotated about \( P(1, 2) \):
- \( x' = 2(1) - (-3) = 2 + 3 = 5 \),
- \( y' = 2(2) - 1 = 4 - 1 = 3 \).

So, the coordinates of \( A' \) are:
\[
A'(5, 3).
\]

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**(b) Find the distance between \( A \) and \( B \).**

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
\]

Substitute \( A(-3, 1) \) and \( B(3, 9) \):
\[
d = \sqrt{(3 - (-3))^2 + (9 - 1)^2} = \sqrt{(3 + 3)^2 + (8)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10.
\]

So, the distance between \( A \) and \( B \) is:
\[
10 \, \text{units}.
\]

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**(c) Yoda claims it is possible to obtain the point \( C(5, 6) \) by rotating the point \( A(-3, 1) \) about \( P(1, 2) \) with some degrees. Do you agree?**

We test whether \( C(5, 6) \) can be obtained by a rotation about \( P(1, 2) \):
1. First, calculate the vector from \( P \) to \( A \):
   \[
   PA = (-3 - 1, 1 - 2) = (-4, -1).
   \]

2. Next, calculate the vector from \( P \) to \( C \):
   \[
   PC = (5 - 1, 6 - 2) = (4, 4).
   \]

3. For \( C(5, 6) \) to be a rotation of \( A(-3, 1) \) about \( P(1, 2) \), the length of \( PA \) and \( PC \) must be equal, and the angle between them must correspond to a standard rotation.

   - Length of \( PA \):
     \[
     |PA| = \sqrt{(-4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}.
     \]

   - Length of \( PC \):
     \[
     |PC| = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32}.
     \]

Since \( |PA| \neq |PC| \), it is **not possible** to obtain \( C(5, 6) \) by rotating \( A(-3, 1) \) about \( P(1, 2) \).

**Conclusion**:
Yoda’s claim is **incorrect**.

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### **Summary of Answers**:
- (a) \( A'(5, 3) \).
- (b) Distance between \( A \) and \( B \) is \( 10 \, \text{units} \).
- (c) It is not possible to obtain \( C(5, 6) \) by rotating \( A(-3, 1) \) about \( P(1, 2) \).

Would you like detailed steps for any part? Let me know!

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### **Further Questions for Practice**:
1. How would the coordinates change if the rotation was \( 90^\circ \) instead of \( 180^\circ \)?
2. How can you find the angle of rotation needed to map \( A \) to \( C \)?
3. What are the coordinates of \( B' \) if \( B(3, 9) \) is rotated \( 180^\circ \) about \( P(1, 2) \)?
4. How would you determine the midpoint of \( AB \)?
5. Can you verify if \( A', B, \) and \( C \) form a specific geometric figure?

**Tip**: Always visualize rotations on a coordinate plane to better understand transformations!

The coordinates of the points A and B are (-3,1) and (3,9) respectively. A is rotated anticlockwise about the point P(1, 2) through 180° to A’. (a) Write down the coordinates of A’. (b) Find the distance between A and B. (c) Yoda claims that it is possible to obtain the point C(5, 6) by rotating the point A about P with some degrees. Do you agree? Explain your answer.

Learn to solve a coordinate geometry problem involving point rotation about another point and finding distances. Includes detailed explanations on calculating rotated coordinates, using the distance formula, and verifying geometric transformations. Suitable for Grades 9-12.

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