To determine how segments [tex]\(XY\)[/tex] and [tex]\(WZ\)[/tex] are related, we need to analyze their slopes by examining the equations of the lines they lie on.
1. Equation of Line for Segment [tex]\(XY\)[/tex]:
- The line equation for segment [tex]\(XY\)[/tex] is given as [tex]\(x - 3y = -12\)[/tex].
- To find the slope, we rewrite it in the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
x - 3y = -12 \implies 3y = x + 12 \implies y = \frac{1}{3}x + 4
\][/tex]
- Here, the slope [tex]\(m_{XY}\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
2. Equation of Line for Segment [tex]\(WZ\)[/tex]:
- The line equation for segment [tex]\(WZ\)[/tex] is given as [tex]\(x - 3y = -6\)[/tex].
- Similarly, we rewrite it in the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
x - 3y = -6 \implies 3y = x + 6 \implies y = \frac{1}{3}x + 2
\][/tex]
- Here, the slope [tex]\(m_{WZ}\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
3. Comparison of Slopes:
- Segments [tex]\(XY\)[/tex] and [tex]\(WZ\)[/tex] have the slopes [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex] respectively.
- Since the slopes are equal, the segments are parallel.
Therefore, the correct statement that proves how segments [tex]\(XY\)[/tex] and [tex]\(WZ\)[/tex] are related is:
- "They have the same slope of [tex]\(\frac{1}{3}\)[/tex] and are, therefore, parallel."
So, the correct statement is:
They have the same slope of [tex]\(\frac{1}{3}\)[/tex] and are, therefore, parallel.
