Finding Equations Of Perpendicular Lines Passing Through A Point
Finding the equation of a line that is perpendicular to a given line and passes through a specific point is a fundamental concept in coordinate geometry. This problem combines the ideas of slope, perpendicularity, and the point-slope form of a linear equation. Let's delve into a step-by-step explanation of how to solve this type of problem, ensuring a clear understanding of the underlying principles.
Perpendicular Lines: Slopes and Relationships
At the heart of this problem lies the concept of perpendicular lines. Perpendicular lines are lines that intersect at a right angle (90 degrees). The relationship between their slopes is crucial for solving this type of problem. If a line has a slope of 'm', then any line perpendicular to it will have a slope of '-1/m'. This is because the product of the slopes of two perpendicular lines is always -1. This inverse relationship is the foundation for finding the equation of the perpendicular line.
To elaborate, let's consider two lines, Line 1 and Line 2, with slopes m1 and m2, respectively. If these lines are perpendicular, then m1 * m2 = -1. This means that if we know the slope of one line, we can easily find the slope of a line perpendicular to it by taking the negative reciprocal. For example, if Line 1 has a slope of 2, then the slope of Line 2 (perpendicular to Line 1) would be -1/2. This negative reciprocal relationship ensures that the lines intersect at a right angle. Understanding this relationship is essential for tackling problems involving perpendicular lines.
Furthermore, the concept of slope itself is a measure of the steepness and direction of a line. It is defined as the change in the y-coordinate divided by the change in the x-coordinate, often referred to as "rise over run." A positive slope indicates that the line is increasing (going upwards) as you move from left to right, while a negative slope indicates that the line is decreasing (going downwards). A slope of zero represents a horizontal line, and an undefined slope represents a vertical line. The slope is a fundamental property of a line that helps us describe its orientation in the coordinate plane. By understanding the relationship between slopes and perpendicularity, we can solve a wide range of geometric problems, including finding equations of lines and determining angles between lines. The concept of perpendicularity extends beyond lines to other geometric shapes and plays a crucial role in various fields, such as engineering, physics, and computer graphics. By mastering these concepts, you can gain a deeper appreciation for the beauty and interconnectedness of mathematics.
Point-Slope Form: Constructing the Equation
After determining the slope of the perpendicular line, the next step is to use the point-slope form of a linear equation. The point-slope form is a powerful tool for constructing the equation of a line when you know a point on the line and its slope. The point-slope form equation is given by: y - y1 = m(x - x1), where (x1, y1) is a known point on the line and 'm' is the slope of the line. This form is particularly useful because it directly incorporates the slope and a specific point, making it easy to find the equation of the line.
In the context of our problem, we are given a point (2, 6) through which the perpendicular line must pass. This point will serve as our (x1, y1) in the point-slope form. We will also have calculated the slope 'm' of the perpendicular line in the previous step. By substituting these values into the point-slope form equation, we can create an equation that represents the line we are seeking. This equation will initially be in point-slope form, but we can then simplify it to slope-intercept form (y = mx + b) or standard form (Ax + By = C), depending on the desired format.
To illustrate, suppose we have a line with a slope of -2 that passes through the point (3, 1). Using the point-slope form, we would substitute m = -2, x1 = 3, and y1 = 1 into the equation: y - 1 = -2(x - 3). This equation represents the line in point-slope form. We can then simplify this equation to slope-intercept form by distributing the -2 and isolating y: y - 1 = -2x + 6, which simplifies to y = -2x + 7. The point-slope form is a versatile tool that allows us to easily construct the equation of a line given its slope and a point on the line, making it a fundamental concept in coordinate geometry. By mastering this form, you can confidently tackle a wide range of problems involving linear equations and their graphs. It provides a direct and intuitive way to represent lines, making it an essential tool for mathematical problem-solving.
Solving for Perpendicular Equations: A Step-by-Step Approach
To address the specific question of finding the equation of a line perpendicular to a given line and passing through the point (2, 6), we need to consider the cases presented: x = 2, x = 6, y = 2, and y = 6. These equations represent vertical and horizontal lines, which have unique properties when it comes to perpendicularity.
Case 1: Line x = 2
The equation x = 2 represents a vertical line. Vertical lines have an undefined slope and run parallel to the y-axis. A line perpendicular to a vertical line is a horizontal line, which has a slope of 0. Since we want the perpendicular line to pass through the point (2, 6), the equation of the horizontal line will be y = 6. This is because all points on this line have a y-coordinate of 6, and the line runs horizontally through the point (2, 6).
Case 2: Line x = 6
Similarly, the equation x = 6 also represents a vertical line. Following the same logic as in Case 1, a line perpendicular to x = 6 will be a horizontal line. Since the perpendicular line must pass through the point (2, 6), its equation will again be y = 6. This horizontal line has a slope of 0 and intersects the y-axis at y = 6. All points on this line have a y-coordinate of 6, ensuring that it passes through the given point (2, 6).
Case 3: Line y = 2
The equation y = 2 represents a horizontal line. Horizontal lines have a slope of 0 and run parallel to the x-axis. A line perpendicular to a horizontal line is a vertical line, which has an undefined slope. To find the equation of the vertical line that passes through the point (2, 6), we consider the x-coordinate of the point. Since all points on a vertical line have the same x-coordinate, the equation of the perpendicular line will be x = 2. This vertical line passes through all points with an x-coordinate of 2, including the given point (2, 6).
Case 4: Line y = 6
Finally, the equation y = 6 also represents a horizontal line. A line perpendicular to y = 6 will be a vertical line. To pass through the point (2, 6), the vertical line must have the equation x = 2. This is because all points on this line have an x-coordinate of 2, and the line runs vertically through the point (2, 6). In summary, when dealing with vertical and horizontal lines, finding the equation of a perpendicular line involves identifying the opposite type of line (horizontal for vertical and vice versa) and using the given point to determine the specific equation. This approach simplifies the process of finding perpendicular lines in these special cases.
Summarizing the Solutions
In conclusion, the equations of the lines perpendicular to the given lines and passing through the point (2, 6) are as follows:
- For x = 2, the perpendicular line is y = 6.
- For x = 6, the perpendicular line is y = 6.
- For y = 2, the perpendicular line is x = 2.
- For y = 6, the perpendicular line is x = 2.
This exercise highlights the importance of understanding the relationship between slopes of perpendicular lines and the application of the point-slope form. By carefully considering the properties of vertical and horizontal lines, we can efficiently determine the equations of perpendicular lines passing through a given point. This concept is fundamental in coordinate geometry and has wide-ranging applications in various fields, including engineering, physics, and computer graphics. Mastering these principles will provide a solid foundation for tackling more complex geometric problems.
By understanding the concepts of slope, perpendicularity, and the point-slope form, you can confidently solve problems involving lines in the coordinate plane. Remember to pay close attention to the properties of vertical and horizontal lines, as they often present special cases that simplify the solution process. With practice and a solid understanding of these fundamental principles, you can master the art of finding equations of perpendicular lines.
