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 $180^\circ$.

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: $A'(5, 3).$

(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: $10 \, \text{units}.$

(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.

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!

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!