Dilations: find the coordinates

Learning Objectives Define dilation, center of dilation, and scale factor. Calculate the coordinates of a dilated point when the center of dilation is the origin (0,0). Calculate the coordinates of a dilated point when the center of dilation is not the origin. Determine the scale factor of a dilation given a pre-image point, an image point, and the center of dilation. Apply the dilation formula to find the coordinates of all vertices of a dilated polygon. Distinguish between an enlargement and a reduction based on the value of the scale factor. Verify that corresponding sides of a pre-image and its dilated image are parallel. Ever wondered how your phone screen zooms in on a photo, making it bigger but keeping the same shape? 🗺️ That's dilation in action! This tutori...

TermDefinitionExample DilationA transformation that produces an image that is the same shape as the original, but is a different size. It is a non-rigid transformation.A triangle with vertices at (1,1), (2,3), and (3,1) is dilated to create a larger triangle with vertices at (2,2), (4,6), and (6,2). Center of DilationThe fixed point in the plane about which all points are expanded or contracted. All lines connecting corresponding points on the pre-image and image pass through this center.If a square is dilated from the origin (0,0), all its vertices will move further away from or closer to (0,0) along straight lines. Scale Factor (k)The ratio of a length on the image to a corresponding length on the pre-image. It determines how much larger or smaller the image will be.A scale factor of k=...

Dilation Centered at the Origin P(x, y) \rightarrow P'(kx, ky) To find the coordinates of an image point P' after a dilation centered at the origin (0,0), multiply each coordinate of the pre-image point P(x,y) by the scale factor k. Dilation Centered at a Point (a, b) P(x, y) \rightarrow P'(k(x-a)+a, k(y-b)+b) To find the coordinates of an image point P' after a dilation centered at a point C(a,b): 1) Find the distance from the center to the point (x-a, y-b). 2) Multiply these distances by the scale factor k. 3) Add the result back to the center's coordinates (a,b). Finding the Scale Factor (k) k = \frac{\text{image coordinate}}{\text{pre-image coordinate}} \quad \text{or} \quad k = \frac{\text{distance from center to image}}{\text{distance fro...