Question

Choose any point a and any point a'. Suppose a' was created through a dilation. State the coordinates for each and explain or show the steps for finding the center of dilation.

Answer 1

To find the center of dilation between points a and a', use the properties of similar triangles or scale factors. Set up a system of equations with the coordinates and solve to find the dilation center. Two pairs of points are needed for a unique solution.

Step-by-step explanation:

To determine the center of dilation between any point a and its image a' after dilation, you can use the properties of similar triangles or the concept of scaling factors. Assume that the coordinates of point a are (x1, y1) and the coordinates of point a' after dilation are (x2, y2). If the scale factor of the dilation is k, then x2 = k * x1 and y2 = k * y1 for a dilation centered at the origin (0,0). However, if the dilation is centered at a different point, say (h, k), the relationship becomes x2 - h = k * (x1 - h) and y2 - k = k * (y1 - k).

To find the center of dilation, we can set up a system of equations based on the relationships between the original points and their corresponding images after dilation. Solving these equations will give us the coordinates of the center of dilation. It is important to have at least two pairs of points and their images to establish a unique solution for the center's coordinates.