Understanding Dilations
Dilation is a transformation that enlarges or reduces a figure based on a fixed point known as the center of dilation. The figure's size changes according to a scale factor, which determines how much larger or smaller the figure will become.
Key Concepts of Dilations
1. Center of Dilation: The fixed point in the plane from which all points of a figure are expanded or contracted.
2. Scale Factor: A number that describes how much the figure will be enlarged or reduced. A scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 reduces it.
3. Similar Figures: Dilated figures are always similar to the original, meaning they have the same shape but different sizes.
How to Perform Dilations
To perform a dilation, follow these steps:
1. Identify the Center of Dilation: Determine the point from which the dilation will occur.
2. Determine the Scale Factor: Decide on the scale factor that will be applied to the figure.
3. Calculate New Coordinates: For each point of the original figure, apply the following formula:
\[
(x', y') = (k \cdot (x - x_c) + x_c, k \cdot (y - y_c) + y_c)
\]
Where:
- \((x', y')\) are the coordinates of the dilated point.
- \(k\) is the scale factor.
- \((x_c, y_c)\) are the coordinates of the center of dilation.
4. Draw the Dilated Figure: Plot the new points to create the dilated figure.
Examples of Dilation Problems
For practice, let's consider some common problems involving dilations.
Example Problem 1
Given: Triangle ABC with vertices A(1, 2), B(3, 4), C(5, 2), center of dilation O(0, 0), and a scale factor of 2.
Steps:
- Apply the dilation formula to each vertex.
- For A(1, 2):
\[
A'(x', y') = (2 \cdot (1 - 0) + 0, 2 \cdot (2 - 0) + 0) = (2, 4)
\]
- For B(3, 4):
\[
B'(x', y') = (2 \cdot (3 - 0) + 0, 2 \cdot (4 - 0) + 0) = (6, 8)
\]
- For C(5, 2):
\[
C'(x', y') = (2 \cdot (5 - 0) + 0, 2 \cdot (2 - 0) + 0) = (10, 4)
\]
Dilated Triangle A'B'C' has vertices A'(2, 4), B'(6, 8), C'(10, 4).
Example Problem 2
Given: Square DEFG with vertices D(-2, -2), E(-2, 2), F(2, 2), G(2, -2), center of dilation O(0, 0), and a scale factor of 0.5.
Steps:
- Apply the dilation formula to each vertex.
- For D(-2, -2):
\[
D'(x', y') = (0.5 \cdot (-2 - 0) + 0, 0.5 \cdot (-2 - 0) + 0) = (-1, -1)
\]
- For E(-2, 2):
\[
E'(x', y') = (0.5 \cdot (-2 - 0) + 0, 0.5 \cdot (2 - 0) + 0) = (-1, 1)
\]
- For F(2, 2):
\[
F'(x', y') = (0.5 \cdot (2 - 0) + 0, 0.5 \cdot (2 - 0) + 0) = (1, 1)
\]
- For G(2, -2):
\[
G'(x', y') = (0.5 \cdot (2 - 0) + 0, 0.5 \cdot (-2 - 0) + 0) = (1, -1)
\]
Dilated Square D'E'F'G' has vertices D'(-1, -1), E'(-1, 1), F'(1, 1), G'(1, -1).
Dilations Practice Answer Key
Here is a practice answer key for common dilation problems:
- Problem 1: Triangle with vertices A(1, 2), B(3, 4), C(5, 2), center O(0, 0), scale factor 2.
- A': (2, 4)
- B': (6, 8)
- C': (10, 4)
- Problem 2: Square with vertices D(-2, -2), E(-2, 2), F(2, 2), G(2, -2), center O(0, 0), scale factor 0.5.
- D': (-1, -1)
- E': (-1, 1)
- F': (1, 1)
- G': (1, -1)
- Problem 3: Rectangle with vertices P(1, 3), Q(4, 3), R(4, 1), S(1, 1), center O(0, 0), scale factor 3.
- P': (3, 9)
- Q': (12, 9)
- R': (12, 3)
- S': (3, 3)
Conclusion
Understanding dilations is a fundamental aspect of geometry that helps students develop critical thinking and problem-solving skills. Mastering this concept requires practice, and having access to a dilations practice answer key can significantly aid in learning. Use the examples and key provided in this article to enhance your understanding of dilations and improve your performance in geometry. Whether you’re a student seeking to grasp the concept or a teacher looking for resources, the knowledge gained from this article will serve as a valuable reference.
Frequently Asked Questions
What is a dilation in geometry?
A dilation is a transformation that alters the size of a figure while maintaining its shape, scaling it up or down from a fixed point called the center of dilation.
How do you determine the scale factor in a dilation?
The scale factor is found by dividing the length of a side of the dilated figure by the corresponding side of the original figure. A scale factor greater than 1 indicates an enlargement, while a factor less than 1 indicates a reduction.
What is the formula for performing a dilation on a point?
The formula for dilating a point (x, y) from a center point (h, k) with a scale factor 'k' is given by: (x', y') = (h + k(x - h), k + k(y - k)).
What role does the center of dilation play in the transformation?
The center of dilation serves as the fixed point from which all points of the figure are scaled. The distance from the center to each point is multiplied by the scale factor.
Can dilations be performed on figures in different quadrants?
Yes, dilations can be performed on figures in any quadrant. The process remains the same, but the coordinates of the points will change according to their distances from the center of dilation.
What happens to the angles of a figure during dilation?
During a dilation, the angles of the figure remain unchanged. Only the lengths of the sides are affected by the scale factor.
How can you check if a dilation has been performed correctly?
You can check the correctness of a dilation by comparing the ratios of the corresponding sides of the original and dilated figures. If the ratios are equal to the scale factor, the dilation is correct.
What is the relationship between the original figure and the dilated figure?
The original figure and the dilated figure are similar, meaning they have the same shape but different sizes. Their corresponding angles are equal, and corresponding sides are proportional.
Where can I find practice problems for dilations?
Practice problems for dilations can be found in geometry textbooks, online educational platforms, or math resource websites that provide worksheets specifically focused on transformations.
