The problem describes two circles on a coordinate plane. The solid circle has center $(5, 4)$ and radius 3. The dashed circle has center $(9, 6)$ and radius 1. We need to determine the transformations needed to move the solid circle exactly onto the dashed circle. This includes translation and dilation. Also, we need to determine if the two original circles are similar.

1. Problem Description

The problem describes two circles on a coordinate plane. The solid circle has center $(5, 4)$ and radius

3. The dashed circle has center $(9, 6)$ and radius

1. We need to determine the transformations needed to move the solid circle exactly onto the dashed circle. This includes translation and dilation. Also, we need to determine if the two original circles are similar.

2. Solution Steps

First, we translate the solid circle so that its center coincides with the center of the dashed circle.

The solid circle's center is at $(5, 4)$ and the dashed circle's center is at $(9, 6)$.

To translate the solid circle to the dashed circle, we need to move the center $(5, 4)$ to $(9, 6)$.

The x-coordinate needs to change from 5 to 9, which is a translation of $9 - 5 = 4$ units to the right.

The y-coordinate needs to change from 4 to 6, which is a translation of $6 - 4 = 2$ units upwards.

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

Next, we need to dilate the solid circle so that its radius matches the radius of the dashed circle.

The solid circle has a radius of 3 and the dashed circle has a radius of

1. To make the solid circle's radius equal to the dashed circle's radius, we need to multiply the radius of the solid circle by a scale factor $k$ such that $3k = 1$. Therefore, $k = \frac{1}{3}$.

So, we dilate the solid circle by a scale factor of $\frac{1}{3}$.

Two circles are always similar. This is because one circle can always be obtained from another circle by a dilation with the center of dilation being the center of the original circle.

3. Final Answer

Translate the solid circle by 4 units to the right and 2 units up. Dilate the solid circle by a scale factor of $\frac{1}{3}$. Yes, the original solid circle and the dashed circle are similar.