The Dilation Equation Calculator helps determine the transformation of geometric figures by applying a scale factor to their coordinates. This tool is useful in geometry, physics, and engineering to analyze the enlargement or reduction of objects while maintaining proportionality. By entering the original coordinates and the scale factor, users can find the new transformed coordinates of a figure.
Formula of Dilation Equation Calculator
The dilation equation is given by:
New Coordinate = Scale Factor × Original Coordinate
For a more detailed calculation:
(New X, New Y) = (Scale Factor × Original X, Scale Factor × Original Y)
where:
- New X, New Y are the coordinates after dilation.
- Scale Factor (k) determines the transformation:
- If k > 1, the figure enlarges.
- If 0 < k < 1, the figure shrinks.
- If k = 1, the figure remains unchanged.
- Original X, Original Y are the coordinates before dilation.
For dilation with a center point (h, k):
(New X, New Y) = ((Original X – h) × Scale Factor + h, (Original Y – k) × Scale Factor + k)
where:
- (h, k) is the center of dilation.
- Original X, Original Y are the starting coordinates.
- Scale Factor (k) determines how much the figure expands or contracts relative to the center.
Dilation Reference Table
This table provides common dilation scale factors and their effects on objects:
| Scale Factor | Transformation Type |
|---|---|
| 0.5 | Shrink by half |
| 1.0 | No change |
| 1.5 | Enlarge by 50% |
| 2.0 | Double in size |
| 3.0 | Triple in size |
Example of Dilation Equation Calculator
A triangle has the following vertices: A(2,3), B(4,5), and C(6,7). If a dilation with a scale factor of 2 is applied with the origin (0,0) as the center, the new coordinates are calculated as follows:
A’ = (2 × 2, 3 × 2) = (4,6)
B’ = (4 × 2, 5 × 2) = (8,10)
C’ = (6 × 2, 7 × 2) = (12,14)
Thus, the dilated triangle has vertices at A'(4,6), B'(8,10), and C'(12,14).
Most Common FAQs
A dilation is a transformation that changes the size of a figure while maintaining its shape and proportionality. It is defined by a scale factor that determines whether the figure enlarges or shrinks.
A negative scale factor not only changes the size of the figure but also reflects it across the center of dilation.
If the center of dilation is at the origin (0,0), all points scale directly from the origin. If the center is elsewhere, the figure transforms relative to that specific point.