Dilations and parallel lines

Learning Objectives Prove that a dilation takes a line not passing through the center of dilation to a parallel line. Describe the effect of a dilation on a line that passes through the center of dilation. Determine the coordinates of the image of a line after a dilation centered at the origin. Verify that two lines are parallel by comparing their slopes. Write the equation of a line after it has been dilated. Apply the properties of similar triangles to explain the relationship between a line and its dilated image. Ever used a projector to display an image on a screen? 📽️ You're performing a dilation! Why do the top and bottom edges of the original image stay parallel to the top and bottom edges on the screen? This tutorial explores the fundamental connection between...

TermDefinitionExample DilationA transformation that enlarges or reduces a figure proportionally from a fixed point called the center of dilation. The shape of the figure is preserved, but its size changes.A triangle with vertices A(1,1), B(3,1), C(1,4) is dilated by a scale factor of 2 from the origin. The image triangle A'B'C' has vertices A'(2,2), B'(6,2), C'(2,8). Center of DilationThe fixed point in a dilation from which all points in the pre-image are scaled to create the image. All rays connecting corresponding points on the pre-image and image pass through this center.If point P is the center of dilation and A' is the image of A, then P, A, and A' are collinear. Scale Factor (k)The ratio of a length on the image to the corresponding length on...

The Dilation and Parallel Lines Theorem A dilation maps a line not passing through the center of dilation to a parallel line. Use this theorem to state that if line L is dilated from center P (where P is not on L), the image L' will be parallel to L. This is the foundational property connecting these two concepts. Dilation of a Line Through the Center A dilation maps a line passing through the center of dilation to the same line. This is the exception to the parallel line theorem. If the line you are dilating contains the center of dilation, the image is not a new parallel line; it is the exact same line as the pre-image. Coordinate Rule for Dilation (Origin-Centered) D_{O,k}(x, y) = (kx, ky) To find the coordinates of an image point after a dilation centered at...