The problem asks us to describe the transformations required to map a solid circle onto a dashed circle. The solid circle has center $(4, 7)$ and radius $1$. The dashed circle has center $(8, 4)$ and radius $3$. We need to determine the translation and dilation required, and whether the two circles are similar.

1. Problem Description

The problem asks us to describe the transformations required to map a solid circle onto a dashed circle. The solid circle has center $(4, 7)$ and radius $1$. The dashed circle has center $(8, 4)$ and radius $3$. We need to determine the translation and dilation required, and whether the two circles are similar.

2. Solution Steps

First, to translate the solid circle's center to the dashed circle's center, we need to move from $(4, 7)$ to $(8, 4)$.

The change in x is $8 - 4 = 4$, and the change in y is $4 - 7 = -3$.

So, we translate the solid circle by $(4, -3)$.

Next, to dilate the solid circle to match the size of the dashed circle, we need to change the radius from $1$ to $3$.

The scale factor for the dilation is the ratio of the new radius to the original radius:

$scale\ factor = \frac{3}{1} = 3$.

Finally, we need to determine if the circles are similar. All circles are similar, since they all have the same shape. Therefore the original solid circle and the dashed circle are similar.

3. Final Answer

Translate the solid circle $(4, -3)$.

Dilate the solid circle by a scale factor of $3$.

Are the original solid circle and the dashed circle similar? Yes.