The lines shown below are parallel. If the green line has a slope of 3, what is the slope of the red line?
The Correct Answer and Explanation is:
If two lines are parallel, their slopes are equal. In this case, you mentioned that the green line has a slope of 3. Therefore, the slope of the red line must also be 3 since it is parallel to the green line.
To understand this concept better, let’s delve into what slope represents and how it applies to parallel lines.
Definition of Slope
The slope of a line is a measure of its steepness and is calculated as the “rise over run.” This means that for every unit of vertical change (rise), there is a corresponding unit of horizontal change (run). Mathematically, the slope ( m ) is expressed as:
[
m = \frac{y_2 – y_1}{x_2 – x_1}
]
where ( (x_1, y_1) ) and ( (x_2, y_2) ) are two points on the line.
Characteristics of Parallel Lines
Parallel lines have the same slope but different y-intercepts. This means they will never intersect, regardless of how far they are extended. If one line has a slope of ( m_1 ) and another line has a slope of ( m_2 ), and if ( m_1 = m_2 ), the lines are parallel. This property is crucial in various fields, including geometry, algebra, and real-world applications such as engineering and architecture.
Application of the Concept
In your case, the green line’s slope being 3 indicates that for every 1 unit you move to the right along the x-axis, you move up 3 units along the y-axis. The red line, being parallel, behaves the same way. Hence, the slope of the red line is also 3. This equivalence in slope ensures that the two lines will maintain a constant distance apart, reflecting the fundamental property of parallelism.
In summary, since the red line is parallel to the green line with a slope of 3, it also shares this slope, confirming that the slope of the red line is indeed 3.