Understanding slopes is essential in various mathematical and real-world applications. In this article, we will focus specifically on perpendicular lines and learn how to determine their slopes. We will explore the concept of perpendicular lines, explain how to calculate their slopes, and provide some examples to solidify your understanding.

What are Perpendicular Lines?

Perpendicular lines are two lines that intersect at a right angle (90 degrees). Visualizing this, it resembles the shape of the letter ‘L’. To determine if two lines are perpendicular, examine their slopes. If the slopes of two lines are negative reciprocals (e.g., 2/3 and -3/2 or -1/4 and 4), they are said to be perpendicular.

Calculating the Slope of a Perpendicular Line:
To calculate the slope of a perpendicular line, follow these steps:
1. Begin by determining the slope of the given line. Let’s call this slope ‘m’.
2. Calculate the negative reciprocal of ‘m’. To do this, flip the fraction upside down and change its sign.
For example, if the slope of the given line is 2/3, the negative reciprocal will be -3/2.

How can I identify whether two lines are perpendicular without calculating the slopes?

The product of the slopes of two perpendicular lines is -1. Therefore, if you are given the slopes of two lines and multiplying them results in -1, these lines are perpendicular.

Example 1: Calculate the slope of a line, perpendicular to a given line with a slope of 3/4.
Solution:
1. The slope of the given line is m = 3/4.
2. We calculate the negative reciprocal:
Negative reciprocal = -4/3.

Can the slope of a perpendicular line be zero?

No, the slope of a perpendicular line cannot be zero. A line with a slope of zero is a horizontal line, and a perpendicular line to it would be a vertical line, whose slope is undefined.

Example 2: Determine the slope of a line that is perpendicular to a horizontal line.
Solution:
Since the slope of a horizontal line is zero, a line that is perpendicular to it will be vertical. The slope of a vertical line is undefined.

Is it possible for parallel lines to be perpendicular?

No, parallel lines cannot be perpendicular to each other. Parallel lines can never intersect, let alone intersect at a right angle.

Example 3: Calculate the slope of a line, perpendicular to a line represented by the equation y = 2x + 5.
Solution:
1. The equation of the given line is y = 2x + 5. Comparing this with the slope-intercept form y = mx + b, we identify that m = 2.
2. Calculating the negative reciprocal:
Negative reciprocal = -1/2.

Understanding perpendicular lines and knowing how to calculate their slopes is an important skill in mathematics. By identifying the slopes of two lines and determining if they are negative reciprocals, you can confirm their perpendicularity. Make use of the examples provided in this article to strengthen your understanding of calculating perpendicular line slopes.

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