To solve this problem, we'll determine the slope and the length of segment \(\overline{AB}\), then find the length of the dilated segment \(\overline{A'B'}\) after a dilation with a scale factor of 3.5.
### Step 1: Determine the Slope (\(m\))
First, we need to find the slope of the line segment \(\overline{AB}\). The coordinates of points \(A\) and \(B\) are \(A(2, 2)\) and \(B(3, 8)\), respectively. The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the given coordinates into the formula:
[tex]\[ m = \frac{8 - 2}{3 - 2} = \frac{6}{1} = 6 \][/tex]
Therefore, the slope of \(\overline{AB}\) is \(6\).
### Step 2: Calculate the Original Length of \(\overline{AB}\)
Next, we use the distance formula to find the length of \(\overline{AB}\). The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substitute the given coordinates into the formula:
[tex]\[ d = \sqrt{(3 - 2)^2 + (8 - 2)^2} \][/tex]
[tex]\[ d = \sqrt{1^2 + 6^2} \][/tex]
[tex]\[ d = \sqrt{1 + 36} \][/tex]
[tex]\[ d = \sqrt{37} \][/tex]
Thus, the original length of \(\overline{AB}\) is \(\sqrt{37}\).
### Step 3: Calculate the Length of \(\overline{A'B'}\) After Dilation
To find the length of \(\overline{A'B'}\) after dilation, we multiply the original length by the scale factor \(3.5\):
[tex]\[ A'B' = 3.5 \times \sqrt{37} \][/tex]
Therefore, the length of \(\overline{A'B'}\) after dilation is \(3.5 \sqrt{37}\).
### Final Answer
Based on our calculations:
- The slope \(m\) is \(6\).
- The length of \(\overline{A'B'}\) after dilation is \(3.5 \sqrt{37}\).
The correct answer is:
[tex]\[ \boxed{C. m = 6, A'B' = 3.5 \sqrt{37}} \][/tex]
