We are given two circles. The solid circle has center $(2, 3)$ and radius $1$. The dashed circle has center $(6, 5)$ and radius $4$. We need to describe the transformations required to move the solid circle exactly onto the dashed circle. This includes a translation and a dilation. We also need to determine if the two circles are similar.

1. Problem Description

We are given two circles. The solid circle has center $(2, 3)$ and radius $1$. The dashed circle has center $(6, 5)$ and radius $4$. We need to describe the transformations required to move the solid circle exactly onto the dashed circle. This includes a translation and a dilation. We also need to determine if the two circles are similar.

2. Solution Steps

First, we need to find the translation vector. The translation vector will move the center of the solid circle to the center of the dashed circle. The center of the solid circle is $(2, 3)$, and the center of the dashed circle is $(6, 5)$.

The horizontal translation is $6 - 2 = 4$ units.

The vertical translation is $5 - 3 = 2$ units.

So, we translate the solid circle by 4 units and by 2 units.

Next, we need to find the scale factor for the dilation. The radius of the solid circle is 1, and the radius of the dashed circle is

4. The scale factor is the ratio of the radii:

Scale factor $= \frac{\text{radius of dashed circle}}{\text{radius of solid circle}} = \frac{4}{1} = 4$.

So, we dilate the solid circle by a scale factor of

4.

Finally, we need to determine if the two circles are similar. Two circles are always similar because any circle can be mapped to another circle by a dilation and a translation. Therefore, the original solid circle and the dashed circle are similar.

3. Final Answer

Translate the solid circle

by 4 unit(s) and

by 2 unit(s).

Dilate the solid circle by a scale factor of

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

Yes