Dilation Calculator
Calculate dilations of points and shapes from any center of dilation with any scale factor. Understand enlargements, reductions, and reflections.
Scale Factor Guide: k > 1 (enlargement), 0 < k < 1 (reduction), k < 0 (reflection + scaling), k = 1 (no change)
Understanding Dilations
What is a Dilation?
A dilation is a transformation that enlarges or reduces a figure by a scale factor relative to a fixed center point. All points move along lines through the center, with distances multiplied by the scale factor.
Dilation Formulas
Dilation from Origin:
(x, y) → (kx, ky)
Dilation from Center (h, k):
1. Translate to origin: (x - h, y - k)
2. Apply scale factor: (k(x - h), k(y - k))
3. Translate back: (k(x - h) + h, k(y - k) + k)
Simplified:
(x, y) → (k(x - h) + h, k(y - k) + k)
Scale Factor Types
k > 1: Enlargement - figure gets larger
Example: k = 2 doubles all distances from center
k = 1: Identity - no change
0 < k < 1: Reduction - figure gets smaller
Example: k = 0.5 halves all distances from center
k < 0: Dilation with reflection through center
Example: k = -2 doubles size and reflects through center
k = 0: All points collapse to the center
Properties of Dilations
- Preserved: Angle measures, shape, orientation (if k > 0), parallelism
- Changed: Size, distances, perimeter, area
- The center of dilation is the only fixed point (doesn't move)
- All points lie on rays from the center
- Distance from center is multiplied by |k|
- Perimeter is multiplied by |k|
- Area is multiplied by k²
- Volume is multiplied by k³
Real-World Applications
- Photography: Zooming in and out, image resizing
- Maps: Scale models and map scales
- Architecture: Creating scale models of buildings
- Computer Graphics: Scaling objects in games and applications
- Manufacturing: Creating scaled prototypes
- Medicine: Medical imaging magnification
Frequently Asked Questions
How do you dilate a point by a scale factor?
From the origin: multiply both coordinates by k. For example, (3, 4) with k = 2 becomes (6, 8). From another center (h, k): use the formula (k(x - h) + h, k(y - k) + k).
What's the difference between enlargement and reduction?
Enlargement occurs when k > 1, making the figure larger. Reduction occurs when 0 < k < 1, making the figure smaller. Both preserve shape and angles, only changing size.
What happens when the scale factor is negative?
A negative scale factor causes the figure to be reflected through the center of dilation, then scaled by the absolute value of k. For example, k = -2 reflects through the center and doubles the size.
How does dilation affect area?
Area is multiplied by the square of the scale factor (k²). If k = 3, the area becomes 9 times larger. If k = 0.5, the area becomes 0.25 times the original (one-quarter). Perimeter is multiplied by |k|.
Are dilated figures similar or congruent?
Dilated figures are similar (same shape, proportional sizes) but not congruent unless k = ±1. Similar figures have equal corresponding angles and proportional corresponding sides.
How do you find the center of dilation?
If you know corresponding points, the center lies on the lines connecting each point to its image. Find where these lines intersect. Alternatively, use the formula: if (x₁, y₁) → (x₂, y₂) with factor k, the center is ((kx₁ - x₂)/(k - 1), (ky₁ - y₂)/(k - 1)).
Can you reverse a dilation?
Yes! The inverse dilation uses the reciprocal scale factor (1/k). If you dilated by k = 3, reverse it by dilating by k = 1/3 from the same center. This returns the figure to its original size.
How are dilations used in similar figures?
All similar figures can be related by a dilation (possibly combined with rigid motions). The scale factor equals the ratio of corresponding side lengths. This relationship is fundamental in geometry and trigonometry.