What is a Geometric Dilation?
A geometric dilation is a transformation that changes the size of a figure while preserving its shape. Every dilation has a center of dilation (a fixed point) and a scale factor (denoted by k) that determines how much the figure is enlarged or reduced.
Key characteristics of dilations:
- The shape of the figure remains unchanged (similar figures)
- All corresponding angles remain equal
- All corresponding sides are proportional by the scale factor k
- The area changes by a factor of k²
Understanding Scale Factors
The scale factor k determines the type of transformation:
Enlargement
When k > 1
The figure becomes larger than the original
Same Size
When k = 1
The figure remains congruent to the original
Reduction
When 0 < k < 1
The figure becomes smaller than the original
Area Changes in Dilations
One of the most important concepts in geometric dilations is how the area changes. When a figure is dilated by a scale factor k, its linear dimensions are multiplied by k, but its area is multiplied by k².
Area scaling formula:
Example: Area Calculation
Consider a rectangle with an original area of 36 square units. If it's dilated with a scale factor of $\frac{1}{3}$:
$\text{New Area} = 36 \times (\frac{1}{3})^2 = 36 \times (\frac{1}{9}) = 4 \text{ square units}$
Finding Scale Factors from Areas
When you know the original and new areas, you can find the scale factor using the relationship:
Scale factor from areas:
Example: Scale Factor Calculation
If a triangle's area changes from 20 cm² to 80 cm², what is the scale factor?
$k = \sqrt{\frac{80}{20}} = \sqrt{4} = 2$
The triangle was enlarged by a scale factor of 2.
Real-World Applications
Geometric dilations have many practical applications:
- Architecture: Scaling blueprints and floor plans
- Photography: Enlarging or reducing images
- Manufacturing: Creating scale models
- Maps: Representing real distances at different scales
- Computer Graphics: Resizing digital images and models
Practice Problem
Problem:
A square garden has an area of 100 m². If it's dilated with a scale factor of 1.5, what will be the new area?
Show Solution
Given: Original area = 100 m², Scale factor k = 1.5
New Area = Original Area × k²
New Area = 100 × (1.5)²
New Area = 100 × 2.25 = 225 m²