Dilations: Scale factor and similar figures

Understanding Dilations 📐

In geometry, a dilation is a transformation that changes the size of a figure. It either enlarges or reduces the figure, but it does not alter its shape. Dilations are defined by two key elements: a center of dilation and a scale factor.

Scale Factor Explained 📏

The scale factor ($k$) determines how much the figure is enlarged or reduced. Here's how it works:

• If $k > 1$, the dilation is an enlargement (the figure gets bigger).
• If $0 < k < 1$, the dilation is a reduction (the figure gets smaller).
• If $k = 1$, the dilation results in the same figure (no change).
• If $k < 0$, the dilation results in an enlargement or reduction and a reflection across the center of dilation.

To perform a dilation, you multiply the coordinates of each point in the original figure by the scale factor. For example, if a point has coordinates $(x, y)$ and the scale factor is $k$, the new coordinates after dilation will be $(kx, ky)$.

Example of Dilation 💡

Let's say we have a triangle with vertices A(1, 1), B(2, 1), and C(1, 2). We want to dilate this triangle by a scale factor of 2, with the origin (0, 0) as the center of dilation.

Here's how we calculate the new coordinates:

• A'(2*1, 2*1) = A'(2, 2)
• B'(2*2, 2*1) = B'(4, 2)
• C'(2*1, 2*2) = C'(2, 4)

The new triangle A'B'C' is an enlargement of the original triangle ABC, and it is similar to the original triangle.

Similar Figures and Dilations 👯

Dilations create similar figures. Similar figures have the same shape but different sizes. This means their corresponding angles are congruent (equal), and their corresponding sides are proportional.

Properties of Similar Figures:

1. Corresponding angles are equal.
2. Corresponding sides are in proportion (the ratio of their lengths is constant).

When a figure is dilated, the resulting figure is always similar to the original figure. The scale factor determines the ratio of the corresponding side lengths.

Code Example: Dilation Transformation 💻

Here's a Python code snippet using the `shapely` library to perform a dilation on a polygon:

from shapely.geometry import Polygon
from shapely.affinity import scale

# Define a polygon
polygon = Polygon([(0, 0), (1, 0), (1, 1), (0, 1)])

# Define the scale factor
scale_factor = 2.0

# Perform dilation with respect to the origin
dilated_polygon = scale(polygon, xfact=scale_factor, yfact=scale_factor, origin=(0, 0))

print(dilated_polygon)
# Output: POLYGON ((0 0, 2 0, 2 2, 0 2, 0 0))

In summary 📝

• Dilation changes the size of a figure using a scale factor.
• A scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 reduces it.
• Dilations create similar figures, preserving angles and proportionality of sides.