Find Equation Of Perpendicular Line Passing Through A Point
Introduction
In the realm of coordinate geometry, understanding the relationships between lines and points is fundamental. One common problem involves finding the equation of a line that is perpendicular to a given line and passes through a specific point. This article delves into the step-by-step process of solving such a problem, providing a clear and comprehensive explanation suitable for students and enthusiasts alike. We will explore the underlying concepts of slope, perpendicular lines, and the point-slope form of a linear equation, ultimately leading to a solution. Let's embark on this mathematical journey to unravel the intricacies of coordinate geometry.
Problem Statement
Consider a coordinate plane where a line passes through two distinct points: (-3, 2) and (0, 1). Additionally, there is a point located at (3, 4). The objective is to determine the equation of the line that satisfies two crucial conditions: it must be perpendicular to the given line (the line passing through (-3, 2) and (0, 1)), and it must pass through the specified point (3, 4). This problem encapsulates key concepts in coordinate geometry, including the calculation of slope, the relationship between slopes of perpendicular lines, and the utilization of point-slope form to derive the equation of a line. Solving this problem requires a systematic approach, employing these concepts to arrive at the final equation. This exercise not only reinforces the understanding of these concepts but also enhances problem-solving skills in a geometric context.
Finding the Slope of the Given Line
To embark on the journey of finding the equation of the perpendicular line, the first critical step is to determine the slope of the given line. This line, as stated in the problem, traverses through two points on the coordinate plane: (-3, 2) and (0, 1). The slope, often denoted as m, is a measure of the steepness and direction of a line. It quantifies the change in the y-coordinate for a unit change in the x-coordinate. Mathematically, the slope is calculated as the ratio of the difference in the y-coordinates to the difference in the x-coordinates between two points on the line. Using the coordinates of the two points, we can apply the slope formula:
m = (y₂ - y₁) / (x₂ - x₁)
Substituting the given coordinates (-3, 2) and (0, 1) into the formula, we get:
m = (1 - 2) / (0 - (-3)) m = -1 / 3
Thus, the slope of the given line is -1/3. This negative slope indicates that the line slopes downward from left to right. The magnitude of the slope (1/3) provides information about the steepness of the line; a smaller magnitude indicates a gentler slope, while a larger magnitude indicates a steeper slope. This calculated slope will be instrumental in determining the slope of the line perpendicular to it, as perpendicular lines have slopes that are negative reciprocals of each other. Understanding the slope of the given line is the foundation for the subsequent steps in solving the problem.
Determining the Slope of the Perpendicular Line
Now that we have successfully calculated the slope of the given line, the next crucial step is to determine the slope of the line that is perpendicular to it. In coordinate geometry, a fundamental property governs the relationship between the slopes of perpendicular lines: they are negative reciprocals of each other. This means that if a line has a slope of m, any line perpendicular to it will have a slope of -1/m. This relationship stems from the geometric definition of perpendicularity, where two lines intersect at a right angle (90 degrees). The negative reciprocal relationship ensures that the lines meet at this precise angle.
Recall that the slope of the given line was found to be -1/3. To find the slope of the perpendicular line, we need to take the negative reciprocal of this value. This involves two operations: first, we take the reciprocal (flipping the fraction), and second, we change the sign. So, the reciprocal of -1/3 is -3/1, which simplifies to -3. Then, we take the negative of -3, which results in 3. Therefore, the slope of the line perpendicular to the given line is 3. This positive slope indicates that the perpendicular line slopes upward from left to right, which is in contrast to the downward slope of the given line. The magnitude of the slope (3) indicates that the perpendicular line is steeper than the given line. This calculated slope will be a key component in constructing the equation of the perpendicular line.
Using the Point-Slope Form
With the slope of the perpendicular line now determined, the next step in finding its equation is to utilize the point-slope form. The point-slope form is a powerful tool in coordinate geometry that allows us to write the equation of a line when we know its slope and a point that it passes through. This form is particularly useful when we have a slope calculated from the slopes of perpendicular lines and a specific point that the line must intersect. The point-slope form is expressed mathematically as:
y - y₁ = m(x - x₁)
where m represents the slope of the line, and (x₁, y₁) are the coordinates of the known point on the line. In our problem, we have already determined that the slope of the perpendicular line is 3. We are also given that this line passes through the point (3, 4). This point provides the values for x₁ and y₁ that we need to plug into the point-slope form. Substituting m = 3, x₁ = 3, and y₁ = 4 into the equation, we get:
y - 4 = 3(x - 3)
This equation is the point-slope form of the equation of the perpendicular line. It directly incorporates the slope and the given point, providing a concise representation of the line's properties. The next step will involve simplifying this equation to obtain the slope-intercept form, which is a more common and easily interpretable form of a linear equation. The use of the point-slope form allows us to transition smoothly from the calculated slope and given point to the equation of the line, bridging the gap between geometric properties and algebraic representation.
Converting to Slope-Intercept Form
Having established the equation of the perpendicular line in point-slope form, the final step is to convert it to the slope-intercept form. The slope-intercept form is a widely recognized and used format for linear equations, as it explicitly reveals the slope and y-intercept of the line. This form is expressed as:
y = mx + b
where m represents the slope of the line, and b represents the y-intercept, which is the point where the line intersects the y-axis. To convert the equation from point-slope form to slope-intercept form, we need to isolate y on one side of the equation. This involves distributing any constants and combining like terms. Recall that the point-slope form of the equation we derived was:
y - 4 = 3(x - 3)
First, we distribute the 3 on the right side of the equation:
y - 4 = 3x - 9
Next, to isolate y, we add 4 to both sides of the equation:
y = 3x - 9 + 4
Combining the constant terms, we get:
y = 3x - 5
This equation is now in slope-intercept form. We can clearly see that the slope of the perpendicular line is 3 (as we calculated earlier), and the y-intercept is -5, meaning the line intersects the y-axis at the point (0, -5). This slope-intercept form provides a complete algebraic representation of the perpendicular line, allowing us to easily visualize and analyze its behavior on the coordinate plane. The conversion to this form solidifies our solution and provides a clear answer to the problem.
Final Answer
In conclusion, after systematically navigating through the problem, we have successfully determined the equation of the line that is perpendicular to the given line and passes through the point (3, 4). The process involved several key steps, each building upon the previous one. We began by calculating the slope of the given line using the coordinates of the two points it passes through. Then, we utilized the property of perpendicular lines to find the slope of the line perpendicular to the given line. Next, we employed the point-slope form to construct the equation of the perpendicular line, incorporating its slope and the given point. Finally, we converted the equation to slope-intercept form to explicitly reveal the slope and y-intercept.
The equation of the line that satisfies the given conditions is:
y = 3x - 5
This equation represents a line with a slope of 3 and a y-intercept of -5. It is perpendicular to the line passing through (-3, 2) and (0, 1), and it passes through the point (3, 4). This solution demonstrates the power of coordinate geometry in solving problems involving lines and points, highlighting the importance of understanding concepts such as slope, perpendicularity, and linear equations. The systematic approach employed in this solution can be applied to a variety of similar problems, reinforcing the fundamental principles of coordinate geometry.
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Find Equation of Perpendicular Line Passing Through a Point
