Dilation
Dilation is a Grade 8 geometry transformation skill in Saxon Math Course 3, Chapter 5, that resizes a figure by a scale factor centered at a given point, producing a similar figure with proportional dimensions. Students perform dilations on the coordinate plane, identify scale factors from image and pre-image measurements, and connect dilation to the study of similar figures and proportional reasoning.
Key Concepts
Property A dilation is a transformation where a figure grows larger or smaller from a fixed point called the center of dilation. To find the new coordinates from the origin, multiply the original coordinates by the scale factor.
Examples Dilating a vertex at $(1, 2)$ with a scale factor of 2 results in a new vertex at $(1 \cdot 2, 2 \cdot 2) = (2, 4)$. A contraction on a vertex at $(6, 2)$ using a scale factor of $\frac{1}{2}$ moves it to $(6 \cdot \frac{1}{2}, 2 \cdot \frac{1}{2}) = (3, 1)$.
Explanation This is geometry's 'zoom' button! A shape gets bigger (dilation) or smaller (contraction) based on a scale factor, expanding from or shrinking toward a center point. The figure's shape remains proportional, but its size changes. A scale factor greater than 1 enlarges, while one less than 1 shrinks.
Common Questions
What is a dilation in Grade 8 geometry?
A dilation is a transformation that produces a figure similar to the original by multiplying all distances from a center point by a scale factor, resulting in an enlarged or reduced image.
How do you perform a dilation on the coordinate plane?
Multiply both coordinates of each point by the scale factor. For example, with scale factor 2, the point (3, 5) maps to (6, 10).
What is the center of dilation?
The center of dilation is the fixed point from which all distances are scaled. When the center is the origin, you simply multiply all coordinates by the scale factor.
How is dilation used to create similar figures?
Dilation preserves the shape and angle measures of a figure but changes its size by the scale factor. The original and image are similar figures with proportional corresponding sides.
Where is dilation taught in Grade 8 Saxon Math?
Dilation is covered in Saxon Math Course 3, Chapter 5: Number and Operations and Algebra.