To determine the relationship between segments [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex], we need to find the slopes of the lines they lie on and see if they have any special geometric relationship, such as being perpendicular or parallel.
First, we begin with the equations of the lines in their given forms and convert them to the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope.
For line [tex]\(AB\)[/tex]:
Given equation: [tex]\(y - 9 = -4(x + 1)\)[/tex]
1. Distribute the [tex]\(-4\)[/tex]:
[tex]\[
y - 9 = -4x - 4
\][/tex]
2. Add 9 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -4x + 5
\][/tex]
Thus, the slope of line [tex]\(AB\)[/tex] (denoted as [tex]\(m_{AB}\)[/tex]) is [tex]\(-4\)[/tex].
For line [tex]\(CD\)[/tex]:
Given equation: [tex]\(y - 6 = \frac{1}{4}(x - 3)\)[/tex]
1. Distribute [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
y - 6 = \frac{1}{4}x - \frac{3}{4}
\][/tex]
2. Add 6 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{1}{4}x + 6 - \frac{3}{4}
\][/tex]
3. Combine constants:
[tex]\[
y = \frac{1}{4}x + \frac{21}{4}
\][/tex]
Thus, the slope of line [tex]\(CD\)[/tex] (denoted as [tex]\(m_{CD}\)[/tex]) is [tex]\(\frac{1}{4}\)[/tex].
Checking the Relationship:
To determine if lines are perpendicular, their slopes need to be opposite reciprocals of each other.
- The opposite reciprocal of [tex]\(-4\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
If [tex]\(m_{AB}\)[/tex] is [tex]\(-4\)[/tex] and [tex]\(m_{CD}\)[/tex] is [tex]\(\frac{1}{4}\)[/tex], then since these slopes are indeed opposite reciprocals, the lines are perpendicular.
Thus, the appropriate statement proving the relationship between segments [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] is:
They are perpendicular because they have slopes that are opposite reciprocals of [tex]\(-4\)[/tex] and [tex]\(\frac{1}{4}\)[/tex].
