Write An Equation Of A Line That Is Perpendicular

Understanding how to write an equation of a line that is perpendicular to another line is a fundamental concept in coordinate geometry. Perpendicular lines are special because they intersect at a right angle (90 degrees), and their slopes have a unique relationship that allows us to construct them precisely.

To start, recall that the slope of a line determines its steepness and direction. If a line has a slope of m, then any line perpendicular to it must have a slope that is the negative reciprocal of m. In other words, if the original line's slope is m, the perpendicular line's slope is -1/m. For example, if a line has a slope of 2, the slope of any line perpendicular to it would be -1/2.

Let's walk through the process step by step. Suppose you are given a line with the equation y = 2x + 3, and you want to find the equation of a line that is perpendicular to it and passes through the point (4, -1).

First, identify the slope of the given line. In this case, the slope is 2. The slope of the perpendicular line will be -1/2, the negative reciprocal of 2.

Next, use the point-slope form of a line, which is y - y₁ = m(x - x₁), where (x₁, y₁) is a point the line passes through and m is the slope. Plugging in the values, you get: y - (-1) = -1/2(x - 4) y + 1 = -1/2x + 2 y = -1/2x + 1

So, the equation of the line perpendicular to y = 2x + 3 and passing through (4, -1) is y = -1/2x + 1.

It's important to remember that horizontal and vertical lines are also perpendicular to each other, but their slopes are special cases: a horizontal line has a slope of 0, and a vertical line has an undefined slope. For example, the line y = 5 is horizontal, and any line of the form x = c (where c is a constant) is vertical and thus perpendicular to y = 5.

In summary, to write an equation of a line that is perpendicular to another line, you need to:

1. Find the slope of the original line.
2. Calculate the negative reciprocal of that slope.
3. Use the point-slope form with a given point (if provided) to find the equation of the perpendicular line.

This method ensures that your new line will always intersect the original line at a right angle, fulfilling the geometric requirement for perpendicularity.