Dilation
Grade 8 math lesson on dilation transformations that enlarge or reduce figures using a center point and scale factor. Students learn to perform dilations on the coordinate plane, find scale factors, and understand how dilation creates similar figures.
Key Concepts
Property A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. A dilation is an 'enlargement,' while a contraction makes the figure smaller. Dilations are similarity transformations.
Examples Dilating a rectangle with vertices $(1,1), (4,1), (4,3), (1,3)$ by a scale factor of 3 from the origin gives new vertices at $(3,3), (12,3), (12,9), (3,9)$. A circle with a radius of 12 units undergoes a contraction with a scale factor of $\frac{1}{4}$. Its new radius is 3 units.
Explanation Think of dilation as using the zoom on your phone's camera. If the scale factor is greater than 1, you're zooming in and the shape gets bigger (enlargement). If the scale factor is between 0 and 1, you're zooming out and it gets smaller (contraction). The shape's proportions stay the same, but its size changes proportionally from a center point.
Common Questions
What is a dilation in math?
A dilation is a transformation that enlarges or reduces a figure proportionally from a center point. The scale factor determines how much the figure is stretched (scale factor > 1) or shrunk (scale factor between 0 and 1).
How do you perform a dilation on the coordinate plane?
To dilate a figure with center at the origin, multiply each vertex coordinate by the scale factor. For scale factor 3, point (2, 4) maps to (6, 12). The figure is the same shape but three times larger.
What is the relationship between dilation and similar figures?
Dilation always produces a similar figure — same shape, different size. The original and dilated figures have equal corresponding angles and proportional corresponding sides, which is the definition of similarity.
What happens when the scale factor of a dilation is 1?
When the scale factor is 1, the dilation produces a congruent figure (same size and shape) at the same location — it is effectively no change. When the scale factor is negative, the figure is also reflected through the center.