To solve for the slope [tex]\( m \)[/tex] and length of [tex]\( \overline{A^{\prime} B^{\prime}} \)[/tex], we first need to determine the slope and length of the line segment [tex]\( \overline{A B} \)[/tex]. The given coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are [tex]\( A(2,2) \)[/tex] and [tex]\( B(3,8) \)[/tex], respectively.
### Step 1: Calculate the Slope
The slope [tex]\( m \)[/tex] of a line segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given coordinates:
[tex]\[ m = \frac{8 - 2}{3 - 2} = \frac{6}{1} = 6 \][/tex]
So, the slope [tex]\( m \)[/tex] of [tex]\( \overline{AB} \)[/tex] is [tex]\( 6 \)[/tex].
### Step 2: Calculate the Length of [tex]\( \overline{AB} \)[/tex]
Using the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the given coordinates:
[tex]\[ d = \sqrt{(3 - 2)^2 + (8 - 2)^2} = \sqrt{1^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37} \][/tex]
The length of [tex]\( \overline{AB} \)[/tex] is [tex]\( \sqrt{37} \)[/tex].
### Step 3: Calculate the Length of [tex]\( \overline{A^{\prime}B^{\prime}} \)[/tex] after Dilation
Given the scale factor of [tex]\( 3.5 \)[/tex], the length of the dilated line segment [tex]\( \overline{A^{\prime}B^{\prime}} \)[/tex] is:
[tex]\[ \text{Length of } \overline{A^{\prime}B^{\prime}} = 3.5 \times \text{Length of } \overline{AB} \][/tex]
[tex]\[ \text{Length of } \overline{A^{\prime}B^{\prime}} = 3.5 \times \sqrt{37} \][/tex]
Thus, the length of [tex]\( \overline{A^{\prime}B^{\prime}} \)[/tex] is [tex]\( 3.5 \sqrt{37} \)[/tex].
### Summary
- Slope [tex]\( m \)[/tex] of [tex]\( \overline{AB} \)[/tex]: [tex]\( 6 \)[/tex]
- Length of [tex]\( \overline{A^{\prime}B^{\prime}} \)[/tex]: [tex]\( 3.5 \sqrt{37} \)[/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{C: m = 6, A^{\prime}B^{\prime} = 3.5 \sqrt{37}} \][/tex]
