Dilations are transformations that change the size of a figure while keeping the shape intact. They are often used in various fields of mathematics and real-world applications. Dilations involve multiplying the coordinates of each point in the figure by a scale factor. This scale factor determines the amount of enlargement or reduction. For example, if a figure is dilated with a scale factor greater than 1, the resulting figure will be larger than the original. Conversely, if the scale factor is between 0 and 1, the figure will be reduced in size.

Dilations can be visualized by picturing a figure being expanded or compressed from a central point called the center of dilation. The center of dilation serves as the anchor point around which the figure is scaled. It can be located anywhere on the plane, including inside or outside of the figure.

Let's see an example. Consider a triangle with vertices A, B, and C. If we dilate this triangle by a scale factor of 2, with the center of dilation at the origin (0,0), the new coordinates of the triangle's vertices will be (2A, 2B, 2C), effectively doubling the lengths of the sides while preserving the angles and ratios.

Dilations are a fundamental concept in geometry and can be applied to solve problems in various mathematical and real-life scenarios. Understanding how dilations affect figures will not only deepen your geometric knowledge but will also expand your problem-solving skills in many areas.