Defining Dilation in Geometry
A dilation is a type of geometric transformation that changes the size of a figure, either enlarging or reducing it, without altering its shape or orientation. Every point on the original figure is moved along a ray from a fixed point called the center of dilation, and the distance from the center is multiplied by a constant factor.
Understanding the Scale Factor
The amount by which a figure is resized in a dilation is determined by the scale factor (k). If the absolute value of k is greater than 1, the dilation is an enlargement. If it is between 0 and 1, it's a reduction. A scale factor of 1 means the figure remains unchanged. The new distance of each point from the center of dilation is k times its original distance.
A Simple Dilation Example
Consider a triangle ABC with vertices A(1,1), B(3,1), and C(2,3). If we dilate this triangle from the origin (0,0) with a scale factor of 2, each coordinate is multiplied by 2. The new vertices would be A'(2,2), B'(6,2), and C'(4,6). The dilated triangle A'B'C' is twice as large as ABC, but maintains its exact shape.
Applications of Dilations
Dilations are fundamental in various fields, including cartography (scaling maps), photography (zooming in or out), engineering (creating scaled models), and computer graphics (resizing images and objects). They are crucial for understanding how objects maintain proportional relationships despite changes in size, a concept widely applied in art and architecture as well.