To determine the relationship between segments [tex]\(XY\)[/tex] and [tex]\(WZ\)[/tex], we need to analyze the slopes of the lines that contain these segments based on their given equations.
1. Convert the equations of the lines to slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
- For the line [tex]\( X-3y=-12 \)[/tex]:
[tex]\[
X - 3y = -12 \\
-3y = -X - 12 \\
y = \frac{1}{3}X + 4
\][/tex]
The slope [tex]\( m \)[/tex] of this line is [tex]\( \frac{1}{3} \)[/tex].
- For the line [tex]\( X-3y=-6 \)[/tex]:
[tex]\[
X - 3y = -6 \\
-3y = -X - 6 \\
y = \frac{1}{3}X + 2
\][/tex]
The slope [tex]\( m \)[/tex] of this line is also [tex]\( \frac{1}{3} \)[/tex].
2. Compare the slopes.
- The slope of the line containing segment [tex]\(XY\)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
- The slope of the line containing segment [tex]\(WZ\)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
3. Determine the relationship.
Since both lines have the same slope of [tex]\( \frac{1}{3} \)[/tex], they are parallel.
Therefore, the correct statement is:
They have the same slope of [tex]\(\frac{1}{3}\)[/tex] and are, therefore, parallel.
