3-5-4 Dilations Explained
Key Concepts of Dilations
Dilations are transformations that change the size of a figure without altering its shape. Key concepts include:
- Center of Dilation: The fixed point from which the dilation is performed.
- Scale Factor: The ratio by which the figure is enlarged or reduced.
- Types of Dilations: Enlargements (scale factor > 1) and reductions (0 < scale factor < 1).
- Properties of Dilations: Angle measures, parallelism, and collinearity are preserved.
1. Center of Dilation
The center of dilation is the point from which the dilation is performed. All points of the original figure are moved along lines passing through this center.
Example:
If the center of dilation is at the origin (0,0), a point (3,4) with a scale factor of 2 will move to (6,8).
2. Scale Factor
The scale factor determines how much the figure is enlarged or reduced. A scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 reduces it.
Example:
For a triangle with vertices at (1,1), (2,1), and (1,3), a scale factor of 3 will transform it to vertices at (3,3), (6,3), and (3,9).
3. Types of Dilations
Dilations can be either enlargements or reductions based on the scale factor:
- Enlargement: Scale factor > 1
- Reduction: 0 < Scale factor < 1
Example:
A square with side length 2 units and a scale factor of 0.5 will reduce to a square with side length 1 unit.
4. Properties of Dilations
Dilations preserve certain properties of the original figure:
- Angle Measures: Angles remain the same.
- Parallelism: Parallel lines remain parallel.
- Collinearity: Points that were collinear remain collinear.
Example:
A triangle with angles 30°, 60°, and 90° will still have these angles after a dilation.
Examples and Analogies
To better understand dilations, consider the following analogy:
Imagine a photograph. When you enlarge it, the picture gets bigger but the proportions and details remain the same. Similarly, when you reduce it, the picture gets smaller but the content remains recognizable. This is what happens in a dilation.
Practical Applications
Understanding dilations is crucial for various real-world applications, such as:
- Computer graphics for resizing images and shapes.
- Architecture for scaling models and blueprints.
- Engineering for designing scaled prototypes.