Dilations: graph the image (Tutorial Only)

Learning Objectives Define dilation, center of dilation, and scale factor. Identify the center of dilation and scale factor from a given transformation. Algebraically determine the coordinates of an image after a dilation centered at the origin. Graph the image of a polygon after a dilation centered at the origin. Graph the image of a polygon after a dilation centered at a point other than the origin. Verify that a dilation preserves angle measure and that corresponding sides are parallel. Prove that the ratio of the lengths of corresponding sides of the image and pre-image is equal to the scale factor. Ever used the 'pinch-to-zoom' feature on your phone to make a photo bigger or smaller? 🤏 You were performing a dilation! This tutorial will guide you through di...

TermDefinitionExample DilationA transformation that produces an image that is the same shape as the original, but is a different size. A dilation stretches (enlarges) or shrinks (reduces) a figure proportionally.A triangle with side lengths 3, 4, 5 is dilated by a scale factor of 2 to create a similar triangle with side lengths 6, 8, 10. 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 the center of dilation.If dilating a square from the origin (0,0), every point on the new square will be on a line connecting the origin to the corresponding point on the original square. Scale Factor (k)The ratio of a length on the image to a corresponding length on the pre-ima...

Dilation Centered at the Origin P(x, y) \rightarrow P'(kx, ky) To find the coordinates of the image P' of a point P(x,y) after a dilation centered at the origin (0,0) with a scale factor k, multiply both the x-coordinate and the y-coordinate by k. Dilation Centered at a Point (a, b) P(x, y) \rightarrow P'(k(x-a)+a, k(y-b)+b) To dilate a point P(x,y) from a center C(a,b), first find the horizontal and vertical distances from the center to the point (x-a, y-b). Multiply these distances by the scale factor k. Finally, add these new distances back to the center's coordinates to find the image point P'. Finding the Scale Factor k = \frac{\text{image length}}{\text{pre-image length}} = \frac{CP'}{CP} The scale factor k is the ratio of the dist...