Math Problem Statement

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

Solution

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

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

Step 1: Determine the Angle of Rotation

From the image, we can see that the quadrilateral ABCDABCD is transformed into A′B′C′D′A'B'C'D', and it appears to be a rotation. The center of this rotation is at (−3,1)(-3, 1).

Using the coordinates of points AA and A′A', we can determine the angle of rotation.

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

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

Step 2: Apply Rotation to Point PP

Once the angle of rotation is determined, we can use the same angle to rotate point PP around the center (−3,1)(-3, 1).

  • The coordinates of point PP are approximately (7,5)(7, 5).

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

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.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Rotation
Transformations

Formulas

Rotation of a point (x, y) around another point (h, k) by an angle θ: x' = h + (x - h) * cos(θ) - (y - k) * sin(θ), y' = k + (x - h) * sin(θ) + (y - k) * cos(θ)

Theorems

Rotation Transformation

Suitable Grade Level

Grades 8-10

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