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Segment [tex]$AB$[/tex] falls on line [tex]$6x + 3y = 9$[/tex]. Segment [tex][tex]$CD$[/tex][/tex] falls on line [tex]$4x + 2y = 8$[/tex]. What is true about segments [tex]$AB$[/tex] and [tex][tex]$CD$[/tex][/tex]?

A. They are parallel because they have the same slope of -2.

B. They are perpendicular because they have slopes that are opposite reciprocals of -2 and [tex]$\frac{1}{2}$[/tex].

C. They are parallel because they have the same slope of 2.

D. They are perpendicular because they have opposite reciprocal slopes 2 and [tex]$-\frac{1}{2}$[/tex].

Sagot :

GinnyAnswer
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.
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