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Parallelogram PQRS with vertices P(-3,-4), Q(0, -3), R(-1,-8), and S(-4,-9): a) translation along the vector (7.0) b) 90° counterclockwise rotation about (3,-2)

Parallelogram PQRS with vertices P(-3,-4), Q(0, -3), R(-1,-8), and S(-4,-9): a) translation along the vector (7.0) b) 90° counterclockwise rotation about (3,-2)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Educational Content: Geometric Transformations** **5. Parallelogram PQRS** **Vertices:** - P(-3, -4) - Q(0, -3) - R(-1, -8) - S(-4, -9) **Transformations:** **a) Translation along the vector (7, 0):** - Translation involves sliding the figure along a vector without rotating or flipping it. After translating each vertex by the vector (7, 0): - New vertices: - P'(4, -4) - Q'(7, -3) - R'(6, -8) - S'(3, -9) **b) 90° counterclockwise rotation about (3, -2):** - Rotation involves turning the figure around a fixed point. This transformation needs calculations to determine new coordinates, which are not provided in the original text. **Diagram Explanation:** - The grid shows the original parallelogram PQRS and its translated version. Each point is shifted as per the vector (7, 0), depicted as arrows from original to new positions. **6. Triangle CDE** **Vertices:** - C(-2, -2) - D(1, -4) - E(3, -4) **Transformations:** **a) Dilation with scale factor of 4 centered at (-1, -3):** - Dilation involves resizing the figure proportionally from a center point. When dilated by a scale factor of 4 about (-1, -3): - New vertex: - C'(3, 1) **b) Reflection in the line y = -x:** - Reflection involves flipping the figure over a line to create a mirror image. **Diagram Explanation:** - The grid depicts triangle CDE and its transformation after dilation. The vertices of the transformed triangle need further calculation for complete coordinates. This content highlights the practical application of translation, rotation, dilation, and reflection in coordinate geometry, crucial for understanding transformations in mathematics.
Transcribed Image Text:**Educational Content: Geometric Transformations** **5. Parallelogram PQRS** **Vertices:** - P(-3, -4) - Q(0, -3) - R(-1, -8) - S(-4, -9) **Transformations:** **a) Translation along the vector (7, 0):** - Translation involves sliding the figure along a vector without rotating or flipping it. After translating each vertex by the vector (7, 0): - New vertices: - P'(4, -4) - Q'(7, -3) - R'(6, -8) - S'(3, -9) **b) 90° counterclockwise rotation about (3, -2):** - Rotation involves turning the figure around a fixed point. This transformation needs calculations to determine new coordinates, which are not provided in the original text. **Diagram Explanation:** - The grid shows the original parallelogram PQRS and its translated version. Each point is shifted as per the vector (7, 0), depicted as arrows from original to new positions. **6. Triangle CDE** **Vertices:** - C(-2, -2) - D(1, -4) - E(3, -4) **Transformations:** **a) Dilation with scale factor of 4 centered at (-1, -3):** - Dilation involves resizing the figure proportionally from a center point. When dilated by a scale factor of 4 about (-1, -3): - New vertex: - C'(3, 1) **b) Reflection in the line y = -x:** - Reflection involves flipping the figure over a line to create a mirror image. **Diagram Explanation:** - The grid depicts triangle CDE and its transformation after dilation. The vertices of the transformed triangle need further calculation for complete coordinates. This content highlights the practical application of translation, rotation, dilation, and reflection in coordinate geometry, crucial for understanding transformations in mathematics.
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