Dilations Worksheet Answers: Your Essential Guide!
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Dilations are an essential concept in geometry, particularly in the study of transformations. Understanding dilations helps students grasp how figures can be enlarged or reduced while maintaining their proportionality. In this article, we will explore the concept of dilations, provide a detailed guide to solving dilation problems, and present answers for common dilation worksheets.
What is Dilation? 🤔
Dilation is a transformation that alters the size of a figure while preserving its shape. It involves a center of dilation and a scale factor that determines how much the figure is enlarged or reduced.
Key Components of Dilation:
- Center of Dilation: A fixed point in the plane from which all points are expanded or contracted.
- Scale Factor (k): A ratio that determines how much the figure is enlarged (k > 1) or reduced (0 < k < 1).
The Process of Dilation
To perform a dilation, follow these steps:
- Identify the center of dilation (O) and the scale factor (k).
- For each point (P) in the figure:
- Draw a line from O to P.
- Measure the distance from O to P.
- Multiply this distance by the scale factor (k) to find the new point (P').
- Repeat for all points in the figure to obtain the new dilated figure.
Example of Dilation
Let's say we have a triangle with vertices A(2, 3), B(4, 5), and C(1, 1), and we want to dilate this triangle from the center O(0, 0) with a scale factor of 2.
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For point A(2, 3):
- The distance from O to A = √((2 - 0)² + (3 - 0)²) = √(4 + 9) = √13.
- New point A' = (2 * 2, 2 * 3) = (4, 6).
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For point B(4, 5):
- New point B' = (2 * 4, 2 * 5) = (8, 10).
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For point C(1, 1):
- New point C' = (2 * 1, 2 * 1) = (2, 2).
The dilated triangle has vertices A'(4, 6), B'(8, 10), and C'(2, 2).
Dilation Worksheet Problems
Here’s a sample worksheet with a few problems related to dilations. Students can try to solve these before looking at the answers:
| Problem # | Description | Scale Factor (k) | Center of Dilation |
|---|---|---|---|
| 1 | Dilate point P(2, 3) | 3 | O(0, 0) |
| 2 | Dilate triangle with vertices A(1, 1), B(2, 2), C(3, 3) | 0.5 | O(0, 0) |
| 3 | Dilate rectangle with corners (1, 1), (1, 2), (2, 1), (2, 2) | 2 | O(0, 0) |
| 4 | Dilate point Q(-4, -5) | 4 | O(0, 0) |
Solving the Problems
Let’s take a look at the solutions to these problems:
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Problem 1:
- Original point: P(2, 3)
- New point: P' = (3 * 2, 3 * 3) = (6, 9)
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Problem 2:
- Original points: A(1, 1), B(2, 2), C(3, 3)
- A' = (0.5 * 1, 0.5 * 1) = (0.5, 0.5)
- B' = (0.5 * 2, 0.5 * 2) = (1, 1)
- C' = (0.5 * 3, 0.5 * 3) = (1.5, 1.5)
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Problem 3:
- Original corners: (1, 1), (1, 2), (2, 1), (2, 2)
- New corners after dilation: (2, 2), (2, 4), (4, 2), (4, 4)
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Problem 4:
- Original point: Q(-4, -5)
- New point: Q' = (4 * -4, 4 * -5) = (-16, -20)
Importance of Dilation in Geometry 📐
Dilation is not just a theoretical concept; it has practical applications in various fields, such as architecture, graphic design, and engineering. By understanding dilations, students can enhance their spatial reasoning skills and apply these principles to real-world problems.
Important Note:
"Dilation maintains the proportional relationships between different parts of figures, which is crucial in creating accurate models and designs."
Conclusion
Understanding dilations is a fundamental part of mastering geometric transformations. By practicing problems and solutions, students can strengthen their skills and apply these concepts to advanced topics in geometry. Whether you're preparing for a test or just looking to improve your understanding of mathematics, mastering dilations will serve you well in your academic journey.