Dilations: find the coordinates

Learning Objectives Define dilation, scale factor, and center of dilation. Identify the pre-image and image in a dilation. Apply the dilation rule $P(x, y) \rightarrow P'(kx, ky)$ for a center of dilation at the origin. Calculate the coordinates of a dilated point or figure given the original coordinates and scale factor. Distinguish between an enlargement and a reduction based on the scale factor. Understand how dilations affect the size of a figure while preserving its shape. Ever wondered how a projector makes a small image huge on a screen, or how a tiny model car can perfectly represent a real one? 📽️ These are all examples of dilations! In this lesson, you'll learn how to mathematically 'stretch' or 'shrink' shapes on a coordinate plane a...

TermDefinitionExample DilationA transformation that changes the size of a figure but not its shape. It either enlarges or reduces the figure.A small square becoming a larger square, or a large circle shrinking to a smaller circle. Scale Factor (k)The ratio by which a figure is enlarged or reduced. It tells you how many times larger or smaller the image is compared to the pre-image.If k=2, the new figure is twice as large. If k=0.5, it's half the size. Center of DilationThe fixed point from which all points on a figure are stretched or shrunk. In this lesson, we will focus on the origin (0,0) as the center.If the origin is the center, all points move directly towards or away from (0,0). Pre-imageThe original figure before a transformation is applied.Triangle ABC before it is dilated....

Dilation from the Origin If a point $P(x, y)$ is dilated from the origin $(0,0)$ by a scale factor $k$, its image $P'$ will have coordinates $P'(kx, ky)$. This rule is used to find the new coordinates of any point after a dilation when the center of dilation is the origin. You simply multiply both the x-coordinate and the y-coordinate by the scale factor. Determining Enlargement or Reduction If $k > 1$, the dilation is an enlargement. If $0 < k < 1$, the dilation is a reduction. This rule helps you predict whether the dilated figure will be larger or smaller than the original figure based on the value of the scale factor $k$.