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The transformation ABCD → A'B'C'D' is a rotation around (-3, 1) by ?° Rotate P around (-3, 1) by the same angle. (You may need to sketch things out on paper.) P' = (?,?)

The image you provided shows a transformation of quadrilateral \( ABCD \) to \( A'B'C'D' \). This transformation is a rotation around the point \( (-3, 1) \), and you're tasked with determining the angle of rotation and then applying that same rotation to the point \( P \).

Let's break this down step-by-step:

### Step 1: Determine the Angle of Rotation
From the image, we can see that the quadrilateral \( ABCD \) is transformed into \( A'B'C'D' \), and it appears to be a rotation. The center of this rotation is at \( (-3, 1) \).

Using the coordinates of points \( A \) and \( A' \), we can determine the angle of rotation.

- Point \( A \) has the coordinates \( (-3, -7) \).
- Point \( A' \) has the coordinates \( (-3, 12) \).
- The center of rotation is \( (-3, 1) \).

The angle of rotation can be calculated by finding the angle between the vectors formed from the center of rotation to \( A \) and from the center of rotation to \( A' \). This involves trigonometry.

### Step 2: Apply Rotation to Point \( P \)
Once the angle of rotation is determined, we can use the same angle to rotate point \( P \) around the center \( (-3, 1) \).

- The coordinates of point \( P \) are approximately \( (7, 5) \).

We will use the formula for rotating a point \( (x, y) \) around another point \( (h, k) \) by an angle \( \theta \):

\[
\begin{aligned}
x' &= h + (x - h) \cdot \cos(\theta) - (y - k) \cdot \sin(\theta), \\
y' &= k + (x - h) \cdot \sin(\theta) + (y - k) \cdot \cos(\theta).
\end{aligned}
\]

### Conclusion:
I'll now calculate the exact angle of rotation and the new coordinates of \( P' \). Let me work that out.The transformation \( ABCD \to A'B'C'D' \) is a rotation by \( 180^\circ \) around the point \( (-3, 1) \).

After rotating point \( P(7, 5) \) by the same angle of \( 180^\circ \) around \( (-3, 1) \), the new coordinates of \( P' \) are \( (-13, -3) \).

Would you like further details or clarification on this process?

Here are 5 follow-up questions related to this topic:
1. How is the formula for rotating a point around another point derived?
2. What are the properties of a \( 180^\circ \) rotation, and how do they affect the geometry of shapes?
3. How would you calculate the image of a point under a \( 90^\circ \) clockwise rotation?
4. Could this method be applied to 3D rotations, and if so, how would it change?
5. How can you verify if a transformation is a rotation versus another type of transformation?

**Tip:** When rotating by \( 180^\circ \), every point will map to the point directly opposite it, preserving distances but flipping positions over the center of rotation.

The transformation ABCD → A'B'C'D' is a rotation around (-3, 1) by ?° Rotate P around (-3, 1) by the same angle. P' = (?,?)

This problem involves rotating a quadrilateral ABCD around the point (-3, 1) by 180°. Learn how to apply the same rotation to another point P and find its new coordinates P'. The problem involves the use of transformation formulas in coordinate geometry, suitable for students in Grades 8-10.

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