Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{array}{l} L_{1}:(3,6),(-6,0) \\ L_{2}:(0,-1),\left(5, \frac{7}{3}\right) \end{array}$$

Step-by-Step Solution

Answer

The lines \(L_{1}\) and \(L_{2}\) are parallel.

Step 1: Calculation of slope for \(L_{1}\)

The slope \(m\) of a line passing through points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) is given by \[ m= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]. So, the slope of \(L_{1}\) which passes through (3,6) and (-6,0) is \[m_{1}= \frac{0-6}{-6-3}= \frac{6}{9}= \frac{2}{3}\].

Step 2: Calculation of slope for \(L_{2}\)

Similarly, the slope of \(L_{2}\) which passes through (0,-1) and \((5, \frac{7}{3})\) is \[m_{2}= \frac{\frac{7}{3} - (-1)}{5-0} = \frac{\frac{10}{3}}{5} = \frac{2}{3}\].

Step 3: Comparison of slopes

Both \(m_{1}\) and \(m_{2}\) are equal to \(\frac{2}{3}\). Therefore, the slopes of \(L_{1}\) and \(L_{2}\) are the same, so the lines are parallel.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when studying coordinate geometry since it describes the steepness and direction of a line. The slope is a measure of how much the line rises or falls vertically for each unit of horizontal movement. Mathematically, slope (\( m \)) is calculated by taking the difference in the y-coordinates (vertical change) and dividing it by the difference in the x-coordinates (horizontal change) of two distinct points on the line. This can be expressed as:
\[ m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \]
When applying this formula, if the result is positive, the line slopes upward as it moves from left to right. Conversely, a negative result indicates a downward slope. Additionally, the greater the absolute value of the slope, the steeper the angle of the line. A slope of zero means the line is horizontal, and vertical lines have an undefined slope since the run (horizontal change) would be zero, leading to division by zero, which is not possible.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, intertwines algebra with geometry to describe locations and shapes using an ordered pair of numbers known as coordinates. These coordinates specify a point's position on a coordinate plane, defined by a horizontal number line (the x-axis) and a vertical number line (the y-axis) intersecting at a point called the origin (0,0).

By using the coordinate system, you can calculate distances, midpoints, slopes, and equations of geometric figures. For instance, when dealing with lines, understanding the concept of slope is key to identifying their orientation and relation to other lines. Additionally, concepts like the distance formula \[ d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}} \] and the midpoint formula \[ M = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right) \] are vital for solving various problems in coordinate geometry.

Comparison of Slopes

In coordinate geometry, comparing the slopes of two lines can inform you about their relationship with one another. Parallel lines have the same slope; thus, when their slopes are equal, they never intersect. Perpendicular lines, in contrast, have slopes that are negative reciprocals of one another (i.e., the product of their slopes is -1).

For example, if a line has a slope of \( m = 2 \), a line perpendicular to it would have a slope of \( m = -\frac{1}{2} \). If two lines have slopes that are neither equal nor negative reciprocals, they are neither parallel nor perpendicular, and they intersect at an angle. This concept is vital for solving geometry problems involving the analysis of line relationships, as illustrated in our textbook exercise where the equality of slopes \( \frac{2}{3} \) indicates that the lines are parallel.