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Quadrilateral WXYZ is on a coordinate plane. Segment XY is on the line [tex]\( x - 3y = -12 \)[/tex], and segment WZ is on the line [tex]\( x - 3y = -6 \)[/tex]. Which statement proves how segments XY and WZ are related?

A. They have opposite reciprocal slopes and are, therefore, perpendicular.
B. They have the same slope and are, therefore, parallel.
C. They have opposite reciprocal slopes and are, therefore, perpendicular.
D. They have the same slope and are, therefore, parallel.

Sagot :

GinnyAnswer
To determine how segments XY and WZ are related, let's analyze the given lines they lie on:

1. Equation of the line for segment XY: [tex]\( x - 3y = -12 \)[/tex]
2. Equation of the line for segment WZ: [tex]\( x - 3y = -6 \)[/tex]

We need to find the slopes of these lines to determine the relationship between the segments.

### Step-by-Step Solution:

1. Find the slope of the line [tex]\( x - 3y = -12 \)[/tex]
- Rewrite the equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]), where [tex]\( m \)[/tex] is the slope:
[tex]\[ x - 3y = -12 \][/tex]
[tex]\[ -3y = -x - 12 \][/tex]
[tex]\[ y = \frac{1}{3}x + 4 \][/tex]
- The slope of this line ([tex]\( m_1 \)[/tex]) is: [tex]\( \frac{1}{3} \)[/tex]

2. Find the slope of the line [tex]\( x - 3y = -6 \)[/tex]
- Again, rewrite the equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[ x - 3y = -6 \][/tex]
[tex]\[ -3y = -x - 6 \][/tex]
[tex]\[ y = \frac{1}{3}x + 2 \][/tex]
- The slope of this line ([tex]\( m_2 \)[/tex]) is: [tex]\( \frac{1}{3} \)[/tex]

### Conclusion:
Since both lines have the same slope ([tex]\( \frac{1}{3} \)[/tex]), the segments XY and WZ are parallel.

### Correct Statement:
They have the same slope of [tex]\( \frac{1}{3} \)[/tex] and are, therefore, parallel.
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