solve this dilation. r is (5,8). center of dilation is (3.5,1). scale factor is 5/3.

To solve this dilation, we will apply the following formula for dilation of a point \( P(x, y) \) with respect to the center of dilation \( C(x_c, y_c) \) and scale factor \( k \):

\[
P' = (x', y') = (x_c + k(x - x_c), y_c + k(y - y_c))
\]

Where:
- \( r(5, 8) \) is the point to be dilated.
- The center of dilation is \( C(3.5, 1) \).
- The scale factor \( k = \frac{5}{3} \).

### Step-by-step calculation:
1. \( x_r = 5 \), \( y_r = 8 \)
2. \( x_c = 3.5 \), \( y_c = 1 \)
3. \( k = \frac{5}{3} \)

For the \( x \)-coordinate:
\[
x' = 3.5 + \frac{5}{3}(5 - 3.5) = 3.5 + \frac{5}{3}(1.5) = 3.5 + \frac{7.5}{3} = 3.5 + 2.5 = 6
\]

For the \( y \)-coordinate:
\[
y' = 1 + \frac{5}{3}(8 - 1) = 1 + \frac{5}{3}(7) = 1 + \frac{35}{3} = 1 + 11.67 = 12.67
\]

### Final dilated point:
\[
P'(x', y') = (6, 12.67)
\]

The dilated point is approximately \( (6, 12.67) \).

Do you have any questions, or would you like more details on the dilation process?

Here are 5 related questions to explore further:
1. What happens if the scale factor is negative?
2. How do you find the center of dilation given the original and dilated points?
3. How does dilation affect distances between points?
4. Can dilation change the orientation of a shape?
5. How does dilation differ from other transformations like translation and rotation?

**Tip:** In dilation, a scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 reduces it.

Learn how to perform the dilation of the point (5, 8) with a center at (3.5, 1) and a scale factor of 5/3. This problem involves key geometric concepts such as dilation, coordinate geometry, and scaling transformations. Suitable for students in Grades 8-10.

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