1. Rotation 180° clockwise about the origin 3 A D 1 J - X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Transcribed Image Text:**Rotation 180° Clockwise About the Origin**
To understand the concept of rotating a shape 180° clockwise about the origin, consider the diagram provided.
The diagram features four quadrants with the origin (0,0) at the central intersection of the x-axis and y-axis. The figure in the diagram is a quadrilateral labeled with points A, B, C, and D. The rotation is centered at the origin, as indicated by the coordinate axes.
Key steps of the process:
1. **Identification of original points**: Identify the coordinates of each vertex of the quadrilateral. Let’s assume the coordinates are as follows for simplicity:
- Point A (a, b)
- Point B (c, d)
- Point C (e, f)
- Point D (g, h)
2. **Applying the rotation rule for 180° clockwise**: When rotating a point 180° clockwise about the origin:
- The new coordinates will be (-x, -y) for each point (x, y).
3. **Calculating the new points**: Apply the rotation to each point:
- Point A' (-a, -b)
- Point B' (-c, -d)
- Point C' (-e, -f)
- Point D' (-g, -h)
4. **Plotting the new points**: Plot these new coordinates on the same graph to visualize the rotated quadrilateral.
The resulting shape will be a reflection of the original quadrilateral across both the x-axis and y-axis. The shape’s orientation remains similar, but its position will be inverted relative to the origin.
**Illustration**:
- Original coordinates:
- Point A is on the second quadrant.
- Point B is on the first quadrant.
- Point C is on the fourth quadrant.
- Point D is on the third quadrant.
After rotating 180° clockwise, each point will be as follows:
- A shifts to the fourth quadrant.
- B shifts to the third quadrant.
- C shifts to the second quadrant.
- D shifts to the first quadrant.
This type of transformation is important in geometry and helps in understanding symmetrical properties and spatial reasoning.
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