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Chapter 3: Problem 56

If a line is vertical, what is true of any line that is perpendicular to it?

Short Answer

Expert verified
A line perpendicular to a vertical line is horizontal with a slope of 0.

Step by step solution

01

Understand Vertical Lines

A vertical line has an undefined slope because it goes straight up and down. The equation of a vertical line is written as \(x = k\), where \(k\) is a constant.
02

Identify a Perpendicular Line

A line that is perpendicular to a vertical line must intersect it at a right angle (90 degrees).
03

Slope of the Perpendicular Line

Since vertical lines have undefined slopes, a line perpendicular to it must have a slope that is the negative reciprocal of an undefined slope, which is 0.
04

Equation of the Perpendicular Line

A line with a slope of 0 is horizontal, and its equation can be written as \(y = c\), where \(c\) is a constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Lines
A vertical line is one of the simplest forms of a straight line in geometry. Unlike other lines, it doesn't have a defined slope. This is because it goes straight up and down, without any horizontal movement. The equation of a vertical line is always given as \( x = k \), where \( k \) represents a constant value. For instance, if \( k = 5 \), the line would pass through every point where the x-coordinate is 5. Vertical lines are very important in geometry because they help us understand undefined slopes and their properties.
Undefined Slope
An undefined slope occurs when a line is vertical. Slope generally measures how steep a line is by comparing its vertical rise to its horizontal run. However, vertical lines don't have any horizontal run because they move straight up or down. This results in a division by zero when calculating the slope, which is undefined mathematically. Understanding the concept of undefined slopes helps in identifying vertical lines and their behaviors in a coordinate system.
Horizontal Lines
Horizontal lines are the exact opposite of vertical lines. They run parallel to the x-axis and do not rise or fall at all. The slope of a horizontal line is 0 because there is no vertical change as you move along the line. The equation of a horizontal line can be written as \( y = c \), where \( c \) is a constant value. For example, if \( c = 2 \), the line passes through every point where the y-coordinate is 2. Horizontal lines are useful in understanding perpendicularity in geometry.
Right Angle
A right angle is an angle of exactly 90 degrees. When two lines intersect to form a right angle, they are said to be perpendicular to each other. In dealing with vertical lines, a perpendicular line to it must be horizontal, thus forming a right angle. This concept of right angles is crucial in geometry and helps in understanding the relationship between intersecting lines. Whenever you see two lines forming a right angle, you can be confident they are perpendicular.

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Most popular questions from this chapter

Do \((3,4)\) and \((4,3)\) correspond to the same point in the plane? Explain.

Plot and label each point in a rectangular coordinate system. $$ (4,0) $$

Plot each set of points, and draw a line through them. Then give the equation of the line. $$ (3,5),(3,0), \text { and }(3,-3) $$

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 3 x=y-9 $$

Plot and label each point in a rectangular coordinate system. $$ (-3,5) $$
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