Equation Of A Perpendicular Line Through A Point A(3,8) And Perpendicular To BC

In the realm of coordinate geometry, one fascinating problem involves determining the equation of a line that passes through a specific point and is perpendicular to a given line segment. Let's explore this concept using a concrete example. Consider triangle $ABC$ defined by the points $A(3,8)$, $B(7,5)$, and $C(2,3)$. Our goal is to find the equation of a line that passes through point $A$ and is perpendicular to the line segment $\overline{BC}$. This seemingly simple problem unveils several fundamental geometric principles and algebraic techniques. We will walk through the necessary steps, providing a clear and concise explanation for each stage, ensuring a comprehensive understanding of the solution.

Geometry is more than just shapes and angles; it’s a language that describes the world around us. Understanding the relationships between lines, points, and figures allows us to solve practical problems and appreciate the elegance of mathematical structures. This exercise will not only provide a solution to the specific problem but also strengthen your understanding of slopes, perpendicular lines, and the point-slope form of a linear equation. This article will delve into the intricacies of finding the equation of a line that passes through a specific point and is perpendicular to a given line segment, offering a comprehensive guide for both students and enthusiasts of mathematics.

The concepts of slope, perpendicularity, and linear equations are crucial in various fields, including engineering, computer graphics, and physics. Mastering these concepts will empower you to tackle more complex problems and appreciate the interconnectedness of mathematical ideas. This problem serves as an excellent example of how geometry and algebra work hand in hand to provide solutions to intricate questions. By carefully applying the principles of coordinate geometry, we can derive the equation of the desired line, solidifying our understanding of these essential mathematical tools. Let’s embark on this journey and uncover the beauty and precision of geometric problem-solving, step by step.

Step 1: Determine the Slope of $\overline{BC}$

To find the equation of a line perpendicular to $\overline{BC}$, our first step is to determine the slope of $\overline{BC}$ itself. The slope of a line segment, often denoted by $m$, is a measure of its steepness and direction. It is defined as the change in the $y$-coordinate divided by the change in the $x$-coordinate between any two points on the line. Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

In our case, we have the coordinates of points $B$ and $C$ as $B(7,5)$ and $C(2,3)$, respectively. We can substitute these values into the slope formula to find the slope of $\overline{BC}$. Let’s designate $B$ as $(x_1, y_1)$ and $C$ as $(x_2, y_2)$. Then, we have:

$$m_{BC} = \frac{3 - 5}{2 - 7} = \frac{-2}{-5} = \frac{2}{5}$$

Thus, the slope of $\overline{BC}$ is $\frac{2}{5}$. This positive slope indicates that the line segment rises from left to right. The slope is a critical piece of information because it allows us to determine the slope of any line perpendicular to $\overline{BC}$. Understanding the concept of slope is fundamental in coordinate geometry, and this calculation forms the basis for finding the perpendicular slope in the next step. This foundation is essential for constructing the equation of the line we seek.

The slope is a fundamental concept in coordinate geometry, representing the steepness and direction of a line. It's crucial for various applications, such as determining parallel and perpendicular lines, analyzing linear relationships, and modeling real-world scenarios. By understanding how to calculate and interpret slope, we gain valuable insights into the behavior of lines and their relationships in the coordinate plane. Now that we've found the slope of $\overline{BC}$, we can proceed to the next crucial step: determining the slope of a line perpendicular to it.

The calculation of the slope $m_{BC}$ is not merely a numerical exercise; it's a geometric insight. The value $\frac{2}{5}$ encapsulates the rate at which the line segment $\overline{BC}$ changes vertically with respect to its horizontal change. This understanding is crucial for visualizing the line's orientation and steepness in the coordinate plane. The ability to accurately calculate and interpret slope is a cornerstone of coordinate geometry, enabling us to solve a wide array of problems involving lines and their relationships. This step sets the stage for the next, which will utilize this slope to find the slope of the perpendicular line, a critical element in defining our target line.

Step 2: Determine the Slope of the Perpendicular Line

Now that we know the slope of $\overline{BC}$ is $\frac{2}{5}$, we can determine the slope of any line perpendicular to it. Two lines are perpendicular if they intersect at a right angle (90 degrees). A fundamental property of perpendicular lines is that their slopes are negative reciprocals of each other. If a line has a slope of $m$, then the slope of a line perpendicular to it is $-\frac{1}{m}$.

Let $m_{BC}$ be the slope of $\overline{BC}$, which we found to be $\frac{2}{5}$. Let $m_{\perp}$ be the slope of the line perpendicular to $\overline{BC}$. Using the negative reciprocal relationship, we have:

$$m_{\perp} = -\frac{1}{m_{BC}} = -\frac{1}{\frac{2}{5}} = -\frac{5}{2}$$

Therefore, the slope of the line perpendicular to $\overline{BC}$ is $-\frac{5}{2}$. This negative slope indicates that the line slopes downwards from left to right, which is expected for a line perpendicular to a line with a positive slope. Understanding this negative reciprocal relationship is essential for solving problems involving perpendicular lines in coordinate geometry. It allows us to connect the slopes of two lines based on their geometric relationship, paving the way for finding equations and analyzing geometric figures.

The concept of negative reciprocals for perpendicular lines is a powerful tool in coordinate geometry. It's not just a mathematical rule but a reflection of the geometric relationship between the lines. The negative sign indicates that one line rises while the other falls, and the reciprocal ensures the lines intersect at a right angle. This principle is widely used in various applications, including computer graphics, where perpendicularity is crucial for creating orthogonal projections and 3D models. The accurate determination of the perpendicular slope is vital for constructing the equation of the line that meets our criteria.

The significance of finding the negative reciprocal slope lies in its ability to define a line that forms a right angle with the given line segment. This geometric constraint is essential in many mathematical and real-world applications. For instance, in architecture and engineering, perpendicular lines are fundamental for structural stability and precise construction. By applying the principle of negative reciprocals, we ensure that our calculated slope corresponds to a line that is truly perpendicular to $\overline{BC}$, thus meeting the requirements of our problem. This step bridges the gap between the given geometric condition and the algebraic representation of the line's slope.

Step 3: Use the Point-Slope Form to Create the Equation

Now that we have the slope of the line perpendicular to $\overline{BC}$ ($m_{\perp} = -\frac{5}{2}$) and a point it passes through, $A(3,8)$, we can use the point-slope form of a linear equation to create the equation of the line. The point-slope form is a convenient way to represent a linear equation when we know a point on the line and its slope. The point-slope form is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. In our case, we have $A(3,8)$ as our point, so $x_1 = 3$ and $y_1 = 8$. We also have the slope $m_{\perp} = -\frac{5}{2}$. Substituting these values into the point-slope form, we get:

$$y - 8 = -\frac{5}{2}(x - 3)$$

This is the equation of the line in point-slope form. However, to express the equation in slope-intercept form ($y = mx + b$), we need to simplify this equation further. This involves distributing the slope and isolating $y$ on one side of the equation. The point-slope form is a powerful tool in coordinate geometry because it allows us to directly construct the equation of a line from its slope and a point it passes through. This step lays the groundwork for converting the equation into the more familiar slope-intercept form, making it easier to analyze and graph.

The point-slope form is particularly useful because it directly incorporates the geometric information we have – a point and a slope – into the equation of the line. This form highlights the relationship between the coordinates of any point on the line and the slope, providing a clear geometric interpretation. It's a flexible form that can be easily transformed into other forms, such as slope-intercept form or standard form, depending on the specific requirements of the problem. The application of the point-slope form here demonstrates its power in bridging the gap between geometric properties and algebraic representation.

Using the point-slope form is a strategic step towards finding the equation of our line. It efficiently utilizes the information we have already gathered: the slope of the perpendicular line and the coordinates of point $A$. This form allows us to construct an equation that satisfies the condition of passing through point $A$ and being perpendicular to $\overline{BC}$. The point-slope form is not just a formula; it’s a representation of the line's characteristics in terms of a specific point and its inclination. The next step will involve transforming this equation into a more familiar form, the slope-intercept form, which will reveal the line's y-intercept and provide further insights into its behavior.

Step 4: Convert to Slope-Intercept Form

To convert the equation from point-slope form to slope-intercept form ($y = mx + b$), we need to distribute the slope and isolate $y$ on one side of the equation. We start with the equation we derived in the previous step:

$$y - 8 = -\frac{5}{2}(x - 3)$$

First, distribute the slope $-\frac{5}{2}$ to both terms inside the parentheses:

$$y - 8 = -\frac{5}{2}x + \frac{15}{2}$$

Next, isolate $y$ by adding 8 to both sides of the equation. To add 8 to $\frac{15}{2}$, we need to express 8 as a fraction with a denominator of 2, which is $\frac{16}{2}$:

y = -\frac{5}{2}x + \frac{15}{2} + 8$ $y = -\frac{5}{2}x + \frac{15}{2} + \frac{16}{2}

Now, add the fractions:

$$y = -\frac{5}{2}x + \frac{31}{2}$$

This is the equation of the line in slope-intercept form. The slope is $-\frac{5}{2}$, which we already knew, and the $y$-intercept is $\frac{31}{2}$. This form makes it easy to visualize the line and identify its key characteristics. Converting to slope-intercept form provides a clear and concise representation of the line, allowing for easy graphing and analysis. This final step completes our journey, providing the equation of the line that satisfies our initial conditions.

Converting the equation to slope-intercept form is a crucial step in making the equation readily interpretable. The slope-intercept form, $y = mx + b$, explicitly displays the slope ($m$) and the y-intercept ($b$) of the line. This form is widely used in mathematics and various applications because it provides a clear understanding of the line's behavior and its intersection with the y-axis. The algebraic manipulations performed in this step demonstrate the importance of precision and attention to detail in mathematical problem-solving.

The transition from point-slope form to slope-intercept form involves algebraic manipulation to reveal the line's y-intercept and provide a more intuitive understanding of its position in the coordinate plane. The slope-intercept form is not only a standard representation but also a practical one, allowing us to quickly sketch the line and compare it with other lines. The final equation, $y = -\frac{5}{2}x + \frac{31}{2}$, is the culmination of our efforts, providing a complete and concise answer to the problem posed.

Final Answer

The equation of the line passing through point $A(3,8)$ and perpendicular to $\overline{BC}$ is:

$$y = -\frac{5}{2}x + \frac{31}{2}$$

Thus, the values for the boxes are:

$y = \boxed{-\frac{5}{2}}x + \boxed{\frac{31}{2}}$

This completes the solution to the problem. We have successfully found the equation of the line that meets the specified criteria by systematically applying the principles of coordinate geometry. The journey involved calculating slopes, understanding perpendicular relationships, and using different forms of linear equations. This problem exemplifies the power of combining geometric insights with algebraic techniques to solve mathematical challenges. This solution not only answers the question but also reinforces the fundamental concepts of coordinate geometry, preparing us for more complex problems in the future.