Dilations & Scale Factors Worksheet Answer Key Explained
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Understanding dilations and scale factors is crucial in the field of geometry, especially when dealing with transformations. When students tackle a worksheet on dilations and scale factors, having a comprehensive answer key is essential for grasping the concepts. In this article, we will break down the key points related to dilations and scale factors, provide explanations for common problems, and present an answer key that clarifies the most commonly asked questions.
What Are Dilations?
Dilations are a type of transformation that produces an image that is the same shape as the original, but is a different size. This transformation can either enlarge or shrink an object, depending on the scale factor used. The original object is often called the "pre-image," and the new object is known as the "image."
Key Characteristics of Dilations
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Center of Dilation: The point from which the dilation occurs. This point can be inside, outside, or on the figure itself.
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Scale Factor: This is a number that scales, or enlarges or reduces, the size of the figure. A scale factor greater than 1 results in an enlargement, while a scale factor less than 1 results in a reduction.
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Proportionality: The sides of the pre-image and image are proportional. If the pre-image sides are represented as ( a ) and the scale factor is ( k ), then the corresponding sides of the image will be ( a \times k ).
Understanding Scale Factors
The scale factor is a critical component of dilations. It dictates how much larger or smaller the image will be compared to the pre-image. Here’s a quick reference:
| Scale Factor (k) | Effect on the Image |
|---|---|
| k > 1 | Enlargement |
| k = 1 | No Change |
| 0 < k < 1 | Reduction |
| k < 0 | Reflection (and enlargement/reduction depending on absolute value) |
Example of Scale Factor in Action
Imagine a triangle with vertices at coordinates ( A(1, 2) ), ( B(3, 4) ), and ( C(5, 6) ). If we apply a dilation with a scale factor of 2 centered at the origin, the new vertices will be calculated as follows:
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For point A:
- New A = (1 * 2, 2 * 2) = (2, 4)
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For point B:
- New B = (3 * 2, 4 * 2) = (6, 8)
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For point C:
- New C = (5 * 2, 6 * 2) = (10, 12)
Thus, the new triangle will be larger and maintain the same shape as the original.
The Dilation Equation
The formula for dilating a point ( P(x, y) ) around the center of dilation ( O(0, 0) ) with a scale factor ( k ) can be expressed as:
[ P' = (k \cdot x, k \cdot y) ]
If the center of dilation is at a different point ( O(h, k) ), the formula adjusts to:
[ P' = (h + k(x - h), k + k(y - k)) ]
This modification ensures that the dilation takes place around the desired point.
Answer Key for Common Problems
When working on a dilations and scale factors worksheet, students may encounter a variety of problem types. Below, we present a sample answer key that provides explanations for the types of questions students might face.
| Problem Number | Description | Scale Factor | Answer | Explanation |
|---|---|---|---|---|
| 1 | Dilation of triangle | 2 | Vertices: A(2, 4), B(6, 8) | Each vertex is multiplied by 2, enlarging the triangle. |
| 2 | Center of dilation at (1,1) | 0.5 | Vertices: A(0, 1), B(1, 2) | Each vertex is adjusted towards (1,1) and scaled down. |
| 3 | Finding new coordinates | -1 | Vertices: A(-1, -2) | Reflection across the origin results in negative coordinates. |
| 4 | Scale factor applied to area | 3 | Area = 9 times original | Area changes by the square of the scale factor (3² = 9). |
| 5 | Identifying scale factor | Unknown | k = 4 | The image is 4 times larger than the original, verified with dimensions. |
Important Note
"When dealing with dilations, it’s crucial to remember that while the size of the figure changes, the angles remain congruent and the shape stays the same."
Tips for Solving Dilation Problems
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Identify the Center of Dilation: Knowing the center helps in calculating the coordinates accurately.
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Use Proportions: Understanding that corresponding sides are proportional can simplify many calculations.
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Practice Different Scale Factors: Familiarity with varying scale factors can prepare you for unexpected changes in size.
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Check Your Work: After calculating, always verify by comparing the original and transformed figures.
Dilations and scale factors form a fundamental part of geometry and are crucial for understanding more complex topics. By using this guide and the answer key provided, students can enhance their grasp of these concepts and approach their worksheets with confidence. Whether it's enlarging shapes or understanding proportionality, the principles of dilation will serve as a valuable tool in mathematical exploration.