To determine the relationship between the segments [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] that lie on the lines [tex]\(2x - 4y = 8\)[/tex] and [tex]\(4x + 2y = 8\)[/tex] respectively, we need to analyze their slopes and intercepts.
1. Determine the Slope of Line [tex]\(2x - 4y = 8\)[/tex]:
- First, rewrite the equation in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
2x - 4y = 8 \implies -4y = -2x + 8 \implies y = \frac{1}{2}x - 2
\][/tex]
- The slope ([tex]\(m\)[/tex]) of this line is [tex]\(\frac{1}{2}\)[/tex].
2. Determine the Slope of Line [tex]\(4x + 2y = 8\)[/tex]:
- Rewrite the equation in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
4x + 2y = 8 \implies 2y = -4x + 8 \implies y = -2x + 4
\][/tex]
- The slope ([tex]\(m\)[/tex]) of this line is [tex]\(-2\)[/tex].
3. Analyzing the Slopes:
- Line [tex]\(AB\)[/tex] has a slope of [tex]\(\frac{1}{2}\)[/tex].
- Line [tex]\(CD\)[/tex] has a slope of [tex]\(-2\)[/tex].
4. Checking for Perpendicularity:
- Lines are perpendicular if their slopes are opposite reciprocals, i.e., the product of their slopes should be [tex]\(-1\)[/tex].
- The product of the slopes is:
[tex]\[
\left(\frac{1}{2}\right) \times (-2) = -1
\][/tex]
- Since the product is [tex]\(-1\)[/tex], the lines are perpendicular.
5. Checking for Coincidence:
- For the lines to be the same, they must have identical slopes and the same [tex]\(y\)[/tex]-intercept.
- The slopes of the lines are different ([tex]\(\frac{1}{2}\)[/tex] vs. [tex]\(-2\)[/tex]), hence the lines are not identical nor do they lie on top of one another.
Based on this analysis, here is the correct statement:
- They are perpendicular because they have slopes that are opposite reciprocals of [tex]\(\frac{1}{2}\)[/tex] and [tex]\(-2\)[/tex].
In conclusion, the correct statement is: "They are perpendicular because they have slopes that are opposite reciprocals of [tex]\(\frac{1}{2}\)[/tex] and [tex]\(-2\)[/tex]."
