To calculate the equation of a line perpendicular to a given line, we need to follow a couple of key steps. The process involves understanding the relationship between the slopes of perpendicular lines and applying it to find the desired equation. Let’s dive into the details.
First, we must recall that the slope of a line is a measure of how steep it is and can be calculated as the ratio of the vertical change (the “rise”) to the horizontal change (the “run”) between any two points on the line. For a line given in the slope-intercept form, y = mx + b, the slope is represented by ’m’. When two lines are perpendicular, their slopes are negative reciprocals of each other. This means if the slope of the first line is m, the slope of a line perpendicular to it would be -1/m.
Step 1: Identify the Slope of the Given Line
Given a line with a slope ’m’, we first need to determine its slope. If the line is given in slope-intercept form (y = mx + b), ’m’ is directly provided. However, if the line is given in another form, such as the standard form (Ax + By = C), we would need to convert it to slope-intercept form to identify ’m’. The conversion involves solving the equation for ‘y’, which would give us y = (-A/B)x + C/B, where -A/B represents the slope ’m’.
Step 2: Calculate the Slope of the Perpendicular Line and Use a Point to Find its Equation
Once we have the slope of the given line, we can easily calculate the slope of the line perpendicular to it by taking its negative reciprocal, which is -1/m. To write the equation of the perpendicular line, we also need a point through which the line passes. If we know a point (x1, y1) that lies on the perpendicular line, we can use the point-slope form of a line’s equation, which is y - y1 = m’(x - x1), where m’ is the slope of the perpendicular line (-1/m) and (x1, y1) is the known point. Plugging in the known slope and point, we can simplify this equation into the slope-intercept form to get the final equation of the perpendicular line.
Example Calculation
Let’s say we have a line with the equation y = 2x + 3, and we want to find the equation of a line perpendicular to it that passes through the point (1, 4). The slope of the given line is 2, so the slope of the perpendicular line is -1⁄2. Using the point-slope form with the point (1, 4) and the slope -1⁄2, we get y - 4 = -1⁄2(x - 1). Simplifying this, we get y - 4 = -1/2x + 1⁄2, which further simplifies to y = -1/2x + 9⁄2. This is the equation of the line perpendicular to the given line and passing through the specified point.
Conclusion
Calculating the equation of a line perpendicular to a given line involves understanding the relationship between their slopes and applying the point-slope form of a line’s equation. By identifying the slope of the given line, calculating the slope of the perpendicular line as its negative reciprocal, and using a known point on the perpendicular line, we can derive its equation. This process is fundamental in geometry and has numerous applications in various fields, including physics, engineering, and computer graphics, where the relationships between lines and their equations are crucial.
