Let's analyze the problem step-by-step:
1. Identify the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( A = (2, 2) \)[/tex]
- [tex]\( B = (3, 8) \)[/tex]
2. Calculate the slope [tex]\( m \)[/tex] of [tex]\(\overline{AB}\)[/tex]:
- The slope formula is [tex]\( m = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex]
- Substituting the coordinates of the points:
[tex]\[
m = \frac{8 - 2}{3 - 2} = \frac{6}{1} = 6
\][/tex]
3. Calculate the original 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 coordinates of the points:
[tex]\[
d = \sqrt{(3 - 2)^2 + (8 - 2)^2} = \sqrt{1^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37}
\][/tex]
4. Determine the new length of [tex]\(\overline{A'B'}\)[/tex] after dilation:
- The dilation scale factor is 3.5.
- The length of [tex]\(\overline{A'B'}\)[/tex] is the original length multiplied by the scale factor:
[tex]\[
L_{A'B'} = 3.5 \times \sqrt{37}
\][/tex]
5. Combine the results:
- The slope [tex]\( m \)[/tex] remains unchanged after dilation.
- The new length of [tex]\(\overline{A'B'}\)[/tex] is [tex]\( 3.5 \sqrt{37} \)[/tex].
Putting it all together, the final values are:
- Slope [tex]\( m = 6 \)[/tex]
- Length of [tex]\(\overline{A'B'} = 3.5 \sqrt{37} \)[/tex]
Therefore, the correct answer is:
B. [tex]\( m = 6 \)[/tex], [tex]\( A'B' = 3.5 \sqrt{37} \)[/tex].
