Introduction to Perpendicular Lines and Slope-Intercept Form

Understanding the relationship between lines, their slopes, and the slope-intercept form is essential for many areas of mathematics, including geometry and calculus. One common task is to find the equation of a line that is perpendicular to a given line and passes through a specific point. In this article, we will walk through the process of finding such a line using a step-by-step approach.

Understanding Slope and Perpendicular Lines

A line's slope is a measure of its steepness and can be calculated using the formula m (y? - y?) / (x? - x?). The slope of a line is denoted by the letter m. For two lines to be perpendicular, their slopes must be negative reciprocals of each other. That is, if the slope of one line is m, the slope of the perpendicular line is -1/m.

Given Line and Its Slope

Consider the line given by the equation y - 3x - 2 0. To find the slope, we need to convert this equation to slope-intercept form, which is y mx b, where m is the slope and b is the y-intercept.

Starting with the given equation:

Move the terms involving x to the other side:

y 3x 2

From this, we can see that the slope m of the given line is -3.

Slope of Perpendicular Line

Since the slope of the required line (the one perpendicular to the given line) must be the negative reciprocal of the slope of the given line, the slope m'1 of the perpendicular line is:

m'1 1/(-3) 1/3

So, the slope of the required line is 1/3.

Equation of the Perpendicular Line

Step 1: Using the Point-Slope Form

The point-slope form of a line's equation is given by:

y - y? m(x - x?)

where (x?, y?) is a point on the line, and m is the slope of the line. Here, we are given the point (2, -4), and we have already determined that the slope of the perpendicular line is 1/3.

Substituting these values into the point-slope form:

y - (-4) (1/3)(x - 2)

which simplifies to:

y 4 (1/3)(x - 2)

Step 2: Simplifying to Slope-Intercept Form

To express this in slope-intercept form, we solve for y:

y 4 (1/3)x - (2/3)

y (1/3)x - (2/3) - 4

y (1/3)x - (2/3) - (12/3)

y (1/3)x - (14/3)

Thus, the equation of the perpendicular line is:

y (1/3)x - (14/3)

Conclusion

In this article, we have demonstrated the process of finding the equation of a line that is perpendicular to a given line and passes through a specific point. We started by converting the given equation to slope-intercept form, determined the slope of the perpendicular line, and then used the point-slope form to find the equation of the desired line. The key concepts we covered include the slope of a line, perpendicular lines, and the use of the slope-intercept form.

Additional Tips

Always start by converting given equations to slope-intercept form to easily identify the slope. Recall that the slope of perpendicular lines is the negative reciprocal. Use the point-slope form to derive the equation of the perpendicular line.

Frequently Asked Questions

What if the given line is vertical?

A vertical line has an undefined slope and a perpendicular line to a vertical line is horizontal, with a slope of 0.

Can a line be perpendicular to itself?

No, a line cannot be perpendicular to itself because the angle between a line and itself is 0°, not 90°.

How can I check if two lines are perpendicular?

Check if the product of their slopes is -1. If it is, the lines are perpendicular.