Dilations on the Coordinate Plane
When the center of dilation is the origin, every coordinate is simply multiplied by : . Non-origin centers require an extra step.
Introduction
Coordinate geometry gives us an algebraic formula for dilations. When the center is at the origin, it's as simple as multiplying. When the center is at some other point, we translate → dilate → translate back.
Past Knowledge
Dilations & scale factor (7.1.3). Coordinate transformations (Unit 4).
Today's Goal
Apply the dilation formula on coordinates from origin and non-origin centers.
Future Success
AA/SSS/SAS Similarity (7.2), similar figure proofs.
Key Concepts
Dilation Centered at the Origin
Simply multiply both coordinates by the scale factor .
Dilation Centered at
Subtract the center, scale, then add the center back. This is the general formula.
Finding the Scale Factor from Coordinates
If the center is the origin: (both ratios must match).
Worked Examples
Origin Dilation, k = 2
Dilate with by centered at the origin.
Origin Dilation, k = ⅓
Dilate by centered at the origin.
Non-Origin Center
Dilate by with center .
Use the formula:
Common Pitfalls
Using Origin Formula for Non-Origin Center
only works when the center is . For center , you must use the general formula.
Forgetting the Center Stays Fixed
The center of dilation maps to itself: . If you plug the center into the formula and get a different point, you made an error.
Real-Life Applications
Digital Zoom
Pinch-to-zoom on a touchscreen is a dilation centered at the pinch point. The math behind it uses the non-origin center formula to scale all pixels relative to where your fingers are.
Image Resizing in Design
When designers resize an image from a corner handle, the opposite corner is the center of dilation. The coordinates of every pixel are recalculated using the scale factor.
Practice Quiz
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