Question

error analysis for the dilation with center (0,0) shown on the graph, your friend says the scale factor is \\(\frac{8}{3}\\). what is the correct scale factor? what mistake did your friend likely make? \\(\\) the correct scale factor is \\(\square\\).

Explanation:

Step1: Recall dilation scale factor rule

The scale factor $k$ for a dilation centered at $(0,0)$ is $\frac{\text{image coordinate}}{\text{pre-image coordinate}}$ for a corresponding point.

Step2: Identify correct ratio direction

Assuming the pre-image point is $(3,0)$ and image is $(8,0)$ (or vice versa, but since the image is smaller/larger: if pre-image is $(8,0)$ and image is $(3,0)$, $k=\frac{3}{8}$; if pre-image is $(3,0)$ and image is $(8,0)$ is wrong, but friend used $\frac{8}{3}$. Correct is $\frac{\text{new}}{\text{original}}$, so if original is 8, new is 3, $k=\frac{3}{8}$.

Step3: Identify friend's mistake

Friend reversed the ratio: used $\frac{\text{pre-image coordinate}}{\text{image coordinate}}$ instead of the correct order.

Answer:

The correct scale factor is $\frac{3}{8}$.
Your friend likely reversed the ratio, using the pre-image length divided by the image length instead of the image length divided by the pre-image length.