Problem 49 - Elementary Algebra Within Reach

Question

In Exercises 46-49, determine whether the lines \(L_{1}\) and \(L_{2}\) that pass through the pairs of points are parallel, perpendicular, or neither. $$ \begin{aligned} &L_{1}:(-10,1),(-7,2) \\ &L_{2}:(5,-2),(6,-5) \end{aligned} $$

Step-by-Step Solution

Answer

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

Step 1: Calculate the slopes of line \(L_{1}\) and \(L_{2}\)

Let's calculate the slope of line \(L_{1}\) with its given two points, (-10,1) and (-7,2). The slope is determined by the change in y divided by the change in x: \[ m_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{2 - 1}{-7 - (-10)} = \frac{1}{3} \]We apply the same formula for \(L_{2}\), with points (5,-2) and (6,-5):\[ m_{2} = \frac{-5 - (-2)}{6 - 5} = -3 \]

Step 2: Compare the slopes

Now, we compare the slopes \(m_{1}\) and \(m_{2}\). As we stated in the analysis, if slopes are equal, lines are parallel, if the product of slopes is -1, lines are perpendicular. Here, \(m_{1} \neq m_{2}\) and \(m_{1} * m_{2} = -1\), so, the lines \(L_{1}\) and \(L_{2}\) are perpendicular.

Key Concepts

Slope of a Line

When we talk about the slope of a line in coordinate geometry, we're referring to a numerical measurement of its steepness, incline, or decline. The slope is denoted by the letter 'm' and it represents how much the 'y' (the vertical coordinate) changes for a unit change in 'x' (the horizontal coordinate).

In mathematical terms, the slope is calculated as the ratio of the difference in the y-coordinates, to the difference in the x-coordinates of two distinct points on the line. Formally, given two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\), the slope \(m\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This concept helps us to understand the direction and the angle at which a line tilts from the horizontal.

Parallel and Perpendicular Lines

Parallel and perpendicular lines have specific properties that are defined by their slopes. Lines are parallel if they have the same slope, meaning they'll never intersect and are always equidistant from each other. For example, if two lines, \(L_a\) and \(L_b\), both have a slope of 2, they are considered parallel.

Perpendicular lines, on the other hand, intersect at a 90-degree angle. The mathematical relationship between the slopes of two perpendicular lines, \(L_a\) and \(L_b\), is such that the product of their slopes is -1. If \(L_a\) has a slope of \(m\), then the slope of \(L_b\) must be \(\frac{-1}{m}\) to be perpendicular. This shows an inverse relationship and ensures the 90-degree intersection.

Improving Exercise Understanding

In some cases, students might struggle to visualize or understand the concept of perpendicularity in terms of slopes. It helps to know that a line that rises from left to right has a positive slope, and one that falls has a negative slope. So two lines, one rising and the other falling, can signal perpendicular lines if the absolute values of their slopes are reciprocals.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This involves algebraic formulas and principles to determine the features and relationships between various geometric shapes on a plane. It typically uses a two-dimensional plane with a horizontal axis (x-axis) and a vertical axis (y-axis), known as the Cartesian coordinate system.

The addition of coordinates to the principles of geometry allows for a more precise and analytical approach to understanding shapes, sizes, and positions, transforming qualitative visual geometry into quantitative analytic geometry. This precise approach enables us to solve problems like finding the distance between points, the area of shapes, and the equations of lines, circles, and other geometric figures.