Determining Perpendicular Lines: An SEO Guide

Understanding the Concept of Perpendicular Lines:

When dealing with lines in mathematics, a crucial concept is that of perpendicular lines. A line that is perpendicular to another line forms a right angle (90 degrees) with it. To find the equation of a line that is perpendicular to a given line and passes through a specific point, we need to understand the relationship between the slopes of the lines.

Step-by-Step Guide

Let's consider the given line y 3x - 2 and find the equation of a line that is perpendicular to it and passes through the point (3, -4).

Step 1: Determine the slope of the original line.

The equation of the line is y 3x - 2. The coefficient of x is the slope, so m1 3.

Step 2: Find the slope of the perpendicular line.

The condition for two lines to be perpendicular is that the product of their slopes is -1. Let m2 be the slope of the line perpendicular to the given line. Therefore, we have:

m1 × m2 -1

Substituting the value of m1 we get:

3 × m2 -1

Solving for m2 we obtain:

m2 -1/3

Step 3: Use the point-slope form of the line equation.

The point-slope form of a line equation is given by y - y1 m(x - x1). Here, x1, y1 is the point the line passes through. In this case, x1 3 and y1 -4.

Substitute the values into the equation:

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

y 4 -1/3(x - 3)

Distribute the slope on the right side:

y 4 -1/3x 1

Isolating y we get:

y -1/3x 1 - 4

y -1/3x - 3

The equation of the line perpendicular to y 3x - 2 and passing through the point (3, -4) is:

boxed{y -1/3x - 3}

Geometric Interpretation

To find the equation of the line graphically, you can start by drawing the original line and then finding the line that has a slope of -1/3 (the negative reciprocal of 3) and passes through the point (3, -4).

The y-intercept of the perpendicular line can be found by substituting the point into the equation y -1/3x b. Solving for b,

-4 -1/3(3) b

-4 -1 b

b -3

Therefore, the equation of the perpendicular line is:

y -1/3x - 3

Conclusion

By understanding the concept of perpendicular lines and using the slope and point-slope formula, we can find the equation of a line that is perpendicular to a given line and passes through a specified point. The process involves determining the slope of the perpendicular line, using the point-slope form, and simplifying the equation.