Dilation
A dilation is a geometric transformation that resizes a figure by a scale factor while preserving its shape and angle measures. To dilate point X(2, 4) by a scale factor of 2 from the origin, multiply each coordinate: X becomes X(4, 8). A scale factor greater than 1 enlarges the figure; a factor between 0 and 1 shrinks it. Covered in Grade 7 Saxon Math Course 2, Chapter 8, dilation is the one transformation that creates similar but not congruent figures, making it fundamental to similarity and proportional reasoning.
Key Concepts
Property A dilation is a transformation that changes the size of a figure while preserving its shape.
Examples Dilating $\triangle XYZ$ with vertex $X(2, 4)$ by a scale factor of 2 from the origin gives $X'(4, 8)$. A square with vertex $A(6, 9)$ is dilated by a scale factor of $\frac{1}{3}$. The new vertex is $A'(2, 3)$. To dilate a point $(x, y)$ by a scale factor $k$, the new point is $(kx, ky)$.
Explanation Dilation is like using a magical photocopier that can enlarge or shrink your shape. The new figure is similar—it has the same angles and proportions—but it's a different size. This is the one transformation that creates a non congruent figure. It’s all about scaling up or down from a central point!
Common Questions
What is a dilation in math?
A dilation is a transformation that scales a figure by a given factor from a fixed center point, changing its size while keeping all angles the same and all sides proportional.
How do you perform a dilation on a coordinate plane?
Multiply both the x and y coordinates of each point by the scale factor. For a scale factor of 3 applied to point (2, 5), the image is (6, 15).
What is a scale factor in a dilation?
The scale factor determines how much the figure is enlarged or reduced. A scale factor greater than 1 enlarges, a factor equal to 1 produces no change, and a factor between 0 and 1 reduces the figure.
How is a dilation different from other transformations like translation or rotation?
Translations, rotations, and reflections preserve size and shape (congruence). A dilation changes size, producing a similar but not congruent figure.
When do 7th graders learn about dilations?
Saxon Math, Course 2, Chapter 8 covers dilations as part of the Grade 7 transformations and geometry unit.
What is the relationship between dilation and similar figures?
A dilation creates a similar figure — one with the same shape and proportional sides but a different size. The ratio of corresponding sides equals the scale factor.