To determine the correct slope and the length of the image segment [tex]$A^{\prime}B^{\prime}$[/tex], let's break down the problem step-by-step.
### 1. Calculate the Slope [tex]\( m \)[/tex]
Given points [tex]\( A(2,2) \)[/tex] and [tex]\( B(3,8) \)[/tex], we can find the slope [tex]\( m \)[/tex] of [tex]\(\overline{AB}\)[/tex] using the slope formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\( (x_1, y_1) = (2, 2) \)[/tex] and [tex]\( (x_2, y_2) = (3, 8) \)[/tex]. Plugging these values in:
[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 Length of [tex]\(\overline{AB}\)[/tex]
Next, we use the distance formula to find the length of [tex]\(\overline{AB}\)[/tex]:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the coordinates of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
d = \sqrt{(3 - 2)^2 + (8 - 2)^2} = \sqrt{1^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37}
\][/tex]
### 3. Calculate the Length of Dilated Segment [tex]\(\overline{A'B'}\)[/tex]
The segment [tex]\(\overline{AB}\)[/tex] is dilated by a factor of [tex]\( 3.5 \)[/tex] with respect to the origin. The length of the dilated segment [tex]\(\overline{A'B'}\)[/tex] is:
[tex]\[
\text{Length of } \overline{A'B'} = 3.5 \times \sqrt{37}
\][/tex]
Thus, the length of [tex]\(\overline{A'B'}\)[/tex] is [tex]\( 3.5 \sqrt{37} \)[/tex].
### Conclusions
- The slope [tex]\( m \)[/tex] of [tex]\(\overline{AB}\)[/tex] is [tex]\( 6 \)[/tex].
- The length of the dilated segment [tex]\(\overline{A'B'}\)[/tex] is [tex]\( 3.5 \sqrt{37} \)[/tex].
### Answer
From the multiple-choice options given:
A. [tex]\( m = 21 \)[/tex], [tex]\( \overline{A'B'} = 3.5 \sqrt{37} \)[/tex]
B. [tex]\( m = 6 \)[/tex], [tex]\( \overline{A'B'} = 3.5 \sqrt{37} \)[/tex]
C. [tex]\( m = 21 \)[/tex], [tex]\( \overline{A'B'} = \sqrt{37} \)[/tex]
D. [tex]\( m = 6 \)[/tex], [tex]\( \overline{A'B'} = \sqrt{37} \)[/tex]
The correct answer is:
B. [tex]\( m = 6 \)[/tex], [tex]\( \overline{A'B'} = 3.5 \sqrt{37} \)[/tex]
