Let's analyze the equations of the lines on which the segments [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] lie:
1. The equation of the line on which segment [tex]\(AB\)[/tex] lies is given by:
[tex]\[ y - 9 = -4(x + 1) \][/tex]
2. The equation of the line on which segment [tex]\(CD\)[/tex] lies is given by:
[tex]\[ y - 6 = \frac{1}{4}(x - 3) \][/tex]
To determine the relationship between these segments, we need to find the slopes of the lines.
For the line [tex]\(y - 9 = -4(x + 1)\)[/tex]:
- Rearrange the equation into slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y - 9 = -4(x + 1) \][/tex]
[tex]\[ y - 9 = -4x - 4 \][/tex]
[tex]\[ y = -4x + 5 \][/tex]
So, the slope [tex]\(m\)[/tex] of this line is [tex]\(-4\)[/tex].
For the line [tex]\(y - 6 = \frac{1}{4}(x - 3)\)[/tex]:
- Rearrange the equation into slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y - 6 = \frac{1}{4}(x - 3) \][/tex]
[tex]\[ y - 6 = \frac{1}{4}x - \frac{3}{4} \][/tex]
[tex]\[ y = \frac{1}{4}x + \frac{21}{4} \][/tex]
So, the slope [tex]\(m\)[/tex] of this line is [tex]\(\frac{1}{4}\)[/tex].
Now, compare the slopes:
- The slope of segment [tex]\(AB\)[/tex] is [tex]\(-4\)[/tex].
- The slope of segment [tex]\(CD\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
To determine if the segments are perpendicular, we need to check if the slopes are negative reciprocals of each other.
- The negative reciprocal of [tex]\(-4\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
Since the slopes [tex]\(-4\)[/tex] and [tex]\(\frac{1}{4}\)[/tex] are negative reciprocals of each other, the lines are perpendicular.
Thus, the correct statement is:
They are perpendicular because they have slopes that are opposite reciprocals of [tex]\(-4\)[/tex] and [tex]\(\frac{1}{4}\)[/tex].
