Question

If two lines are perpendicular, what is true of their slopes?

Short answer

The slopes of two perpendicular lines are negative reciprocals of each other.

Step-by-step solution

Understanding Perpendicular Lines

When we say that two lines are perpendicular, we mean they form a right angle (90 degrees) when they intersect. This specific spatial orientation has implications for the slopes of the lines.

Identifying the Slope Relationship

For two lines to be perpendicular, the product of their slopes must be -1. If one line has a slope of \( m_1 \), the other line must have a slope of \( m_2 \) such that \( m_1 \times m_2 = -1 \).

Expression in Terms of Slopes

In explicit terms, the relationship signifies that the slope of one line is the negative reciprocal of the other. Hence, if \( m_1 \) is the slope of the first line, the slope of the second line \( m_2 \) must be \( m_2 = -\frac{1}{m_1} \).

Verification of Perpendicularity

We can verify this rule with examples: if one line's slope is 2 (\( m_1 = 2 \)), then for the lines to be perpendicular, the second line's slope must be \( -\frac{1}{2} \) (\( m_2 = -\frac{1}{2} \)). Checking, \( 2 \times -\frac{1}{2} = -1 \), confirming perpendicularly.

Key concepts

Understanding Slope Relationship in Perpendicular Lines

When dealing with perpendicular lines, one of the essential concepts to grasp is their slope relationship. Slopes measure the steepness and direction of a line. They are a fundamental part of linear equations and line discussions. In coordinate geometry, slopes are calculated as the ratio of the vertical change to the horizontal change between two points on the line.

Understanding the slope relationship between two perpendicular lines means recognizing how these slopes interact to create a right angle. This perpendicularity means they must relate in a special, predictable way.

In essence, if you find that the slopes of two intersecting lines result in a product of -1, these lines aren't just any two intersecting lines—they are perpendicular, forming a perfect 90-degree angle at their meeting point. This mathematical condition helps us quickly identify perpendicular lines just by looking at how their slopes interplay.

Negative Reciprocals and Their Role

To delve deeper into the subject of perpendicular lines, we must explore the concept of negative reciprocals. A reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 5 is \( \frac{1}{5} \). A negative reciprocal takes this one step further by changing the sign. Therefore, the negative reciprocal of 5 is \( -\frac{1}{5} \).

So, how does this relate to perpendicular lines? When two lines are perpendicular, the slope of one line is the negative reciprocal of the other. Suppose line A has a slope \( m_1 \). The slope of a line that is perpendicular to line A, which we'll call line B, is necessarily \( -\frac{1}{m_1} \).

• This mathematical relationship between slopes and negative reciprocals guarantees that when two lines intersect, they form a right angle only if they follow this rule.
• It's a straightforward yet powerful concept that eliminates the need for visual clues of perpendicularity, relying purely on numerical relationships.

Product of Slopes Equals Negative One

Another pivotal aspect of perpendicular lines is understanding the product of slopes. This principle summarizes why slopes as negative reciprocals translate into perpendicularity.

When you multiply the slopes of two perpendicular lines, the result must always be -1. Let's say you have two lines with slopes \( m_1 \) and \( m_2 \). These lines are perpendicular if and only if their product is \( m_1 \times m_2 = -1 \).

• Imagine a line with slope 3. A perpendicular line's slope would need to be \(-\frac{1}{3}\). Multiply them: \(3 \times -\frac{1}{3} = -1\), fulfilling this perpendicular condition.

This concept offers a convenient verification method—no geometrical drawing is necessary. Simply checking the slopes ensures that the lines meet with precise perpendicularity.