Problem 81 - College Algebra and Calculus: An Applied Approach

Question

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(-2,-1),(1,5) ; L_{2}:(1,3),(5,-5)\)

Step-by-Step Solution

Answer

The lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular.

Step 1: Find the slopes of the lines

We can use the formula for the slope, which is \(m = (y_2 - y_1) / (x_2 - x_1)\). The slope of \(L_{1}\) is \(m_1 = (5 - -1) / (1 - -2) = 6 / 3 = 2\). The slope of \(L_{2}\) is \(m_2 = (-5 - 3) / (5 - 1) = -8 / 4 = -2\)

Step 2: Check if the slopes are equal

If the slopes are equal, then the lines are parallel. In this case, \(m_1 = 2\) and \(m_2 = -2\) are not equal, so the lines are not parallel.

Step 3: Check if the product of the slopes is -1

If the product of the slopes is -1, the lines are perpendicular. In this case, \(m_1 * m_2 = 2 * -2 = -4\), which is not equal to -1, so the lines are not perpendicular.

Key Concepts

Slope Calculation

In geometry, the slope of a line is a measure of how steep the line is. It's often represented by the letter \(m\). To calculate the slope of a line that goes through two points, we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula tells us how much the \(y\)-coordinate changes for a given change in the \(x\)-coordinate. To find the slope, simply take the difference between the second and first \(y\)-values and divide by the difference between the respective \(x\)-values.
For example, in the case of line \(L_1\), the points are \((-2, -1)\) and \((1, 5)\). The change in \(y\) is \(5 - (-1) = 6\), and the change in \(x\) is \(1 - (-2) = 3\), so the slope \(m_1 = \frac{6}{3} = 2\).
Calculating the slope is crucial since it helps in understanding the behavior and relationships between different lines, like whether they're parallel or perpendicular.

Parallel Lines

Parallel lines are lines in a plane that never meet. They move in the same direction and have the same angle of steepness. For two lines to be parallel, they must have identical slopes. In simpler terms, if two lines are parallel, their slope values \(m_1\) and \(m_2\) will be the same.
In our exercise, line \(L_1\) has a slope of 2, and line \(L_2\) has a slope of -2. Since 2 is not equal to -2, these lines are not parallel.
Parallel lines are a key concept in geometry, especially when dealing with grids or when calculating angles between intersecting lines. Understanding this concept can help you identify parallel lines quickly in various math problems and real-world applications.

Perpendicular Lines

Perpendicular lines are two lines which intersect at a right angle, forming an angle of 90 degrees. In terms of their slopes, two lines are perpendicular if the product of their slopes equals -1. This means that the slope of one line is the negative reciprocal of the slope of the other.
For example, if one line has a slope \(m_1 = 2\), the line perpendicular to it must have a slope \(m_2 = -\frac{1}{2}\). To verify if two lines are perpendicular, simply multiply their slopes. If the result is -1, then the lines are perpendicular.
In our example, line \(L_1\) has a slope of 2, and line \(L_2\) has a slope of -2. The product \(2 \times -2 == -4\), which is not equal to -1, thus the lines are not perpendicular.
Perpendicular lines are essential in geometry because they help form rectangles and squares and are crucial in various design and engineering applications. Knowing about perpendicular relationships enables problem-solving for angles and intersection points effectively.