Question

Slopes of Parallel and Perpendicular Lines Find the slopes of the lines parallel to, and perpendicular to, each line with the given slope. $$m=-5.372$$

Short answer

The slope of the parallel line is \(m_{parallel} = -5.372\) and the slope of the perpendicular line is approximately \(m_{perpendicular} = \frac{1}{5.372}\).

Step-by-step solution

Parallel Line Slope

The slope of a line parallel to a given line is the same as the slope of the given line. Since the slope of the given line is \(m=-5.372\), any line parallel to this line will have the same slope, \(m_{parallel} = -5.372\).

Perpendicular Line Slope

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. To find the negative reciprocal, divide 1 by the given slope and change the sign. The slope of the line perpendicular to \(m=-5.372\) is \(m_{perpendicular} = \frac{1}{5.372}\), and then change the sign to get the final perpendicular slope \(m_{perpendicular} = \frac{1}{5.372}\). Since the slope was negative, the perpendicular slope will be positive.

Key concepts

Parallel Line Slope

Understanding the slope of a parallel line is a fundamental concept in geometry. Simply put, parallel lines have identical slopes. This means that if you have a line with a slope of \( m = -5.372 \), every line that is parallel to it will also have a slope of \( m_{parallel} = -5.372 \).

Imagine two roads running alongside each other with no point of intersection; these roads are akin to parallel lines in that they remain the same distance apart indefinitely. In the language of mathematics, their consistency in direction is represented by their unchanged slope values.

It is crucial to memorize this property when dealing with linear equations and geometric figures, specifically when predicting the paths of lines on a graph. If you're graphing linear equations, identifying parallel lines can be as simple as comparing their slope values.

Perpendicular Line Slope

A line perpendicular to another exhibits an inverse relationship in terms of their slopes. The slope of a perpendicular line is the negative reciprocal of the original line's slope. To elucidate, the original line with slope \( m = -5.372 \) inversely flips to \( m_{perpendicular} = \frac{1}{5.372} \) and the sign changes to positive.

This flip and sign change are akin to taking a sharp turn from a given direction – it's a complete but proportionate change in direction. Whenever you find the slope of a line and need to determine its perpendicular counterpart, remember to flip the slope and change the sign.

For visualization purposes, if one line is ascending to the right, its perpendicular line will descend to the right. This property is immensely valuable in not just algebraic calculations but also in various fields that deal with right angles, such as architecture and engineering.

Negative Reciprocal

The term negative reciprocal is pivotal when dealing with perpendicular lines. It refers to taking the opposite reciprocal of a given number or slope. When the slope of one line is \( m = -5.372 \), the process to find its negative reciprocal (and thus the slope of a perpendicular line) is a two-step task:

1. Take the reciprocal of the given slope: \( \frac{1}{-5.372} \).
2. Reverse the sign: From negative to positive, resulting in the slope \( m_{perpendicular} = \frac{1}{5.372} \).

This operation ensures that the product of the slopes of two perpendicular lines is -1, which is a defining characteristic of perpendicularly intersecting lines. It's a handy trick to remember for quickly solving problems involving perpendicular lines in geometry and coordinate algebra.

Linear Equations

Linear equations form the basis for lines on a graph. They are expressions that meet two criteria – they represent straight lines, and their highest power of any variable is one. An example of a linear equation is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

When working with linear equations, understanding slopes is essential. Parallel lines will have equations with the same slope \( m \) value, and the equation of a line perpendicular to these will have a slope that is the negative reciprocal of \( m \) from the original line's equation. Any homework or real-life problems involving roads, beams, or even market trends often demand a grasp of linear equations and their slopes for an accurate analysis.

This knowledge comes in handy in various applications such as determining the trajectory of objects, understanding economic trends, or even setting up computer algorithms that deal with linear data patterns.