To determine the relationship between segments [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex], we need to find the slopes of the lines they fall on. First, we will convert the equations of the lines into slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
Step 1: Find the slope of line [tex]\( AB \)[/tex]
The equation of line [tex]\( AB \)[/tex] is:
[tex]\[ 6x + 3y = 9 \][/tex]
First, we solve for [tex]\( y \)[/tex]:
[tex]\[ 3y = -6x + 9 \][/tex]
[tex]\[ y = -2x + 3 \][/tex]
From this equation, we see that the slope [tex]\( m \)[/tex] of line [tex]\( AB \)[/tex] is [tex]\( -2 \)[/tex].
Step 2: Find the slope of line [tex]\( CD \)[/tex]
The equation of line [tex]\( CD \)[/tex] is:
[tex]\[ 4x + 2y = 8 \][/tex]
Again, we solve for [tex]\( y \)[/tex]:
[tex]\[ 2y = -4x + 8 \][/tex]
[tex]\[ y = -2x + 4 \][/tex]
From this equation, we find that the slope [tex]\( m \)[/tex] of line [tex]\( CD \)[/tex] is also [tex]\( -2 \)[/tex].
Step 3: Determine the relationship between the segments
Since both lines have the same slope of [tex]\( -2 \)[/tex], and parallel lines have equal slopes, we can conclude that the segments [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex] are parallel.
Thus, the correct answer is:
a) They are parallel because they have the same slope of -2.
