Question

which motion rule describes the dilation shown below, with the origin as the center of dilation? circle i is the pre - image and circle ii is the image. (x, y)→(5x, 5y) (x, y)→(8x, 8y) (x, y)→(3/8 x, 3/8 y) (x, y)→(8/3 x, 8/3 y)

Explanation:

Step1: Find radii of circles

Radius of Circle I is 2 (counting grid - squares from center to edge), radius of Circle II is 8.

Step2: Calculate scale - factor

Scale - factor $k=\frac{\text{radius of image}}{\text{radius of pre - image}}=\frac{8}{2} = 4$. In a dilation centered at the origin $(x,y)\to(kx,ky)$. Here $k = 4=\frac{8}{2}$, so the rule is $(x,y)\to(4x,4y)$. But we can also note that if we consider the general form of dilation rule. If we assume the rule is $(x,y)\to(ax,ay)$. By comparing the growth of coordinates from pre - image to image. We see that for a point on the pre - image and corresponding point on the image, the ratio of coordinates of image to pre - image gives the scale factor. Since the scale factor from Circle I to Circle II is 4, and we can rewrite 4 as $\frac{8}{2}$. The correct rule for dilation is $(x,y)\to(\frac{8}{2}x,\frac{8}{2}y)$. Among the given options, the closest is $(x,y)\to(\frac{8}{3}x,\frac{8}{3}y)$ as there might be some mis - reading of grid values or approximation in options. But if we assume correct values based on grid, the rule for dilation of a point $(x,y)$ with center of dilation at the origin and scale factor $k$ is $(x,y)\to(kx,ky)$ and here $k = 4$. However, if we consider the options and the concept of dilation rule, we know that the general form of dilation centered at origin is $(x,y)\to(kx,ky)$. The ratio of the size of the image circle to the pre - image circle gives the scale factor.

Answer:

$(x,y)\to(\frac{8}{2}x,\frac{8}{2}y)$ (but if choosing from given options, $(x,y)\to(\frac{8}{3}x,\frac{8}{3}y)$ is the closest)