Problem 33 - Calculus with Concepts in Calculus

Question

Decide which pairs of lines are parallel, which are perpendicular, and which are neither. For any pair that is not parallel, find the point of intersection. \(x=2\) and \(x=-5\)

Step-by-Step Solution

Answer

The lines are parallel and do not intersect.

Step 1: Identify the Form of Each Line

The first line is given as \( x = 2 \), which is a vertical line. The second line is \( x = -5 \), another vertical line. Vertical lines are parallel to each other because they have the same orientation and never intersect.

Step 2: Determine if Lines are Parallel or Perpendicular

Vertical lines like \( x = 2 \) and \( x = -5 \) are parallel as they have no slope value (slope is undefined). Therefore, they cannot be perpendicular to each other, as perpendicular lines need to have slopes that are negative reciprocals of each other.

Step 3: Analyze Intersection

Since both lines are vertical and parallel, they do not intersect at any point. Thus, there is no intersection point to find in this scenario.

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never meet. They remain a constant distance apart and do not intersect no matter how far they are extended. In geometry, parallel lines have the following characteristics:

• Same Orientation: They have the same direction or orientation. For example, vertical lines like those in this exercise (\(x = 2\) and \(x = -5\)) run parallel because they both go straight up and down.
• No Intersection: Since parallel lines never meet, they do not have an intersection point. This is like train tracks that never cross.
• Equal Slopes: In cases where the slope can be defined, parallel lines will have equal slopes. However, vertical lines have undefined slopes, yet they are considered parallel to each other because both lines are the same type (vertical).

Understanding parallel lines is crucial in solving geometric problems. If two lines in a coordinate plane maintain the same angle with respect to an axis and never intersect, they are parallel.

Perpendicular Lines

Perpendicular lines intersect at a right angle, which is 90 degrees. Here are some essential points about perpendicular lines:

• Right Angle: When two lines intersect to form a right angle, they are perpendicular. This geometric property is key when constructing buildings and other structures to ensure effective support and balance.
• Negative Reciprocal Slopes: In a coordinate plane, lines are perpendicular when their slopes are negative reciprocals of one another. For instance, the slope of one line is \(m\), then the slope of a line perpendicular to it would be \(-\frac{1}{m}\).
• Distinct from Parallel Lines: Unlike parallel lines, perpendicular lines must intersect. In the exercise described, \(x = 2\) and \(x = -5\) are vertical lines, hence they cannot be perpendicular as they do not intersect at a right angle or any angle.

Recognizing perpendicular lines helps in solving problems related to angles, determining shapes' properties, and creating axes of symmetry.

Intersection of Lines

The intersection of lines occurs when two or more lines cross at one point. Here’s what you should understand about intersection points:

• Unique Intersection Point: When two lines intersect, they meet at exactly one location called the point of intersection. This point represents the solution to a system of linear equations represented by those lines.
• No Intersection for Parallel Lines: As seen in the given exercise, the lines \(x = 2\) and \(x = -5\) do not intersect as they are parallel. Therefore, there is no point of intersection to determine.
• Handling Vertical and Horizontal Lines: For vertical lines, like in the exercise, intersections occur at distinct x-values. If a vertical and horizontal line intersect, it's straightforward since their equations directly contribute to finding the intersection.

Understanding line intersections is critical in determining solutions for equations, analyzing graphically represented data, and solving geometric problems involving shapes and spaces.