Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. $$(-4,-12)\( and \)(0,-4) ;(0,-5)\( and \)(2,-4)$$

Step-by-Step Solution

Answer

The lines formed by the pairs of points are neither parallel nor perpendicular.

Step 1: Determine the slopes

First, the slopes for the pairs of points need to be calculated. The formula for slope \( m \) between two points \( (x1, y1) \) and \( (x2, y2) \) is given by \( m = (y2 - y1) / (x2 - x1) \). For the first pair of points (-4,-12) and (0,-4), the slope is \( m1 = (-4 - (-12)) / (0 - (-4)) = 2 \).For the second pair of points (0,-5) and (2,-4), the slope is \( m2 = (-4 - (-5)) / (2 - 0) = 0.5 \).

Step 2: Compare the slopes

Since that the slopes from Step 1 are \( m1 = 2 \) and \( m2 = 0.5 \), these do not match the conditions for either parallel or perpendicular lines. Parallel lines would have the same slope and perpendicular lines would have slopes that are negative reciprocals (one is the negative inverse of the other).

Step 3: Conclusion

Therefore, the lines formed by these pairs of points are neither parallel nor perpendicular.

Key Concepts

Understanding the Slope

In geometry, the concept of a slope is crucial for analyzing the direction and steepness of a line on a graph. Simply put, the slope measures how much a line rises or falls as it moves from left to right. The formula for slope \( m \) is:

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

where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
The slope tells us about the line's direction:

• A positive slope means the line rises.
• A negative slope means the line falls.
• A slope of zero indicates a horizontal line.
• An undefined slope shows a vertical line.

This understanding is not only foundational for working with linear equations but is also essential for determining relationships between multiple lines.

Parallel Lines and Their Slopes

Parallel lines are lines in a plane that never intersect. Think of them as train tracks that always run side by side. For lines to be parallel, they must have identical slopes.
This means if you calculate the slope of two lines and they are the same, the lines are parallel.
For example, if Line 1 has a slope of 2 and Line 2 also has a slope of 2, these lines are parallel.
Parallel lines in coordinate geometry are efficient to work with because their equations remain consistent in form.

Perpendicular Lines and Negative Reciprocals

Perpendicular lines intersect at a right angle, forming a perfect "L" shape. If you're considering whether two lines are perpendicular, their slopes are key. They must be negative reciprocals of each other.
Here's what that means:

• If one line has a slope \( m \), the slope of the perpendicular line is \(-1/m\).
• For example, if Line 1 has a slope of \( 2 \), the slope of a line perpendicular to it would be \(-\frac{1}{2} \).

This concept is frequently used to solve problems involving right angles and to determine orthogonal directions in various applications of geometry.

Introduction to Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic methods to study geometric problems.
This field allows you to represent geometric figures on the coordinate plane, which consists of an x-axis and a y-axis.
In coordinate geometry, you deal with points, lines, and shapes using numbers and formulas.

• Lines are expressed in algebraic terms using equations like \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.
• Understanding how to plot points and draw lines is fundamental for solving questions involving linear relationships and interactions.

The intersection, slopes, and distances can be calculated using this brilliant merger of geometry and algebra, making it easier to visualize and solve complex geometrical problems.