(X, Y)
The Dilation Rule Calculator helps users determine the new coordinates of a point after applying dilation. Dilation is a transformation that enlarges or shrinks a figure based on a scale factor while maintaining its shape. This calculator is useful for students, teachers, and professionals working with geometry, graphics, or design.
By using this tool, users can quickly determine how a shape changes when scaled up or down, making it easier to solve geometric problems, model real-world scaling scenarios, or understand proportional transformations.
Formula of Dilation Rule Calculator
The formula for dilation is:
(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 shape enlarges.
- If 0 < k < 1, the shape shrinks.
- If k = 1, the shape remains unchanged.
- Original X, Original Y are the pre-dilation coordinates.
If the dilation has a center at (h, k), the formula adjusts as follows:
(New X, New Y) = ((Original X – h) × Scale Factor + h, (Original Y – k) × Scale Factor + k)
where:
- (h, k) is the center of dilation.
This formula ensures that the figure is proportionally resized while maintaining its orientation relative to the center.
General Dilation Rule Table
The table below provides common dilation transformations based on different scale factors.
| Scale Factor (k) | Effect on Shape | Example Transformation (Original (X, Y) = (4, 6)) |
|---|---|---|
| 2 | Enlargement | (8, 12) |
| 0.5 | Reduction | (2, 3) |
| 1 | No Change | (4, 6) |
| -1 | Reflection & Same Size | (-4, -6) |
| -2 | Reflection & Enlargement | (-8, -12) |
This table helps users quickly understand how dilation affects a shape without performing calculations manually.
Example of Dilation Rule Calculator
Suppose a point (5, 3) is dilated with a scale factor of 2.
Using the formula:
(New X, New Y) = (Scale Factor × Original X, Scale Factor × Original Y)
= (2 × 5, 2 × 3)
= (10, 6)
Thus, the new coordinates after dilation are (10, 6).
If the dilation is centered at (2, 1), then:
(New X, New Y) = ((5 – 2) × 2 + 2, (3 – 1) × 2 + 1)
= (3 × 2 + 2, 2 × 2 + 1)
= (8, 5)
In this case, the new coordinates after dilation from center (2,1) are (8, 5).
Most Common FAQs
A negative scale factor not only changes the size of the shape but also reflects it across the center of dilation.
The center of dilation is the fixed point from which all points expand or contract. If not specified, it is usually the origin (0,0).
No, dilation only resizes the shape proportionally. The angles and relative distances remain the same.