To determine the slope and the length of the dilated segment [tex]\(\overline{A' B'}\)[/tex] for the given points [tex]\( A(2,2) \)[/tex] and [tex]\( B(3,8) \)[/tex], we need to follow these steps:
1. Calculate the slope [tex]\( m \)[/tex] of [tex]\(\overline{AB}\)[/tex]:
The formula for the slope between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting [tex]\( A(2, 2) \)[/tex] and [tex]\( B(3, 8) \)[/tex] into the formula:
[tex]\[
m = \frac{8 - 2}{3 - 2} = \frac{6}{1} = 6
\][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\( 6 \)[/tex].
2. Calculate the original length of [tex]\(\overline{AB}\)[/tex]:
The distance formula between points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting [tex]\( A(2, 2) \)[/tex] and [tex]\( B(3, 8) \)[/tex]:
[tex]\[
d_{AB} = \sqrt{(3 - 2)^2 + (8 - 2)^2} = \sqrt{1^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37}
\][/tex]
3. Dilate [tex]\(\overline{AB}\)[/tex] by scale factor 3.5:
When a segment is dilated by a scale factor [tex]\(k\)[/tex], the length of the new segment becomes [tex]\(k\)[/tex] times the original length.
Here, the scale factor [tex]\(k\)[/tex] is [tex]\(3.5\)[/tex]:
[tex]\[
d_{A'B'} = 3.5 \times d_{AB} = 3.5 \times \sqrt{37}
\][/tex]
Thus, the slope [tex]\( m \)[/tex] remains unchanged because dilation preserves the slopes of lines. The length of the dilated segment [tex]\(\overline{A'B'}\)[/tex] is [tex]\( 3.5 \sqrt{37} \)[/tex].
Putting it all together:
- The slope [tex]\( m \)[/tex] is [tex]\( 6 \)[/tex].
- The length of [tex]\( \overline{A'B'} \)[/tex] is [tex]\( 3.5 \sqrt{37} \)[/tex].
Therefore, the correct answer is:
A. [tex]\(m=6\)[/tex], [tex]\(A^{\prime}B^{\prime}=3.5 \sqrt{37}\)[/tex]
