Finding The Equation Of Line BC A Geometry Problem Solved

In the realm of coordinate geometry, understanding the relationships between points and lines is fundamental. This article delves into a problem involving vectors, right angles, and the determination of a line's equation. Our focus is on finding the equation of the line $\overrightarrow{BC}$ given that $\overrightarrow{AB}$ and $\overrightarrow{BC}$ form a right angle at point $B$, with the coordinates of $A$ and $B$ provided. This exploration will not only reinforce key geometric concepts but also demonstrate the practical application of these concepts in solving analytical problems. By meticulously dissecting the problem, we will navigate through the intricacies of slopes, perpendicularity, and the formulation of linear equations, providing a comprehensive understanding that will enhance your problem-solving prowess in coordinate geometry.

Problem Statement

Let's begin by restating the problem clearly: We are given that vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$ form a right angle at point $B$. The coordinates of point $A$ are $(-3, -1)$, and the coordinates of point $B$ are $(4, 4)$. Our objective is to determine the equation of the line $\overrightarrow{BC}$. This problem elegantly combines the concepts of vectors, perpendicularity, and linear equations, requiring a systematic approach to unravel its solution. To tackle this, we will first ascertain the slope of the line $\overrightarrow{AB}$, which will then allow us to deduce the slope of the perpendicular line $\overrightarrow{BC}$. With the slope of $\overrightarrow{BC}$ and the coordinates of point $B$, we can then construct the equation of the line in point-slope form, subsequently converting it to the standard form. This methodical breakdown will ensure that we not only arrive at the correct answer but also gain a deeper understanding of the underlying geometric principles at play. As we proceed, each step will be carefully elucidated, making the solution accessible and insightful for readers of all backgrounds. The ultimate goal is to empower you with the ability to confidently tackle similar problems, fostering a robust understanding of coordinate geometry.

Understanding the Concepts

Before diving into the solution, let's clarify the key concepts involved. First, a right angle implies that the two lines forming the angle are perpendicular. In coordinate geometry, two lines are perpendicular if the product of their slopes is -1. This is a crucial property that we will utilize to find the slope of $\overrightarrow{BC}$. Secondly, the slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $(y_2 - y_1) / (x_2 - x_1)$. This formula allows us to quantify the steepness and direction of a line. The slope of $\overrightarrow{AB}$ will be the foundation for determining the slope of $\overrightarrow{BC}$. Thirdly, the equation of a line can be expressed in several forms, including the slope-intercept form $(y = mx + b)$, where $m$ is the slope and $b$ is the y-intercept, and the point-slope form $(y - y_1 = m(x - x_1))$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line. We will use the point-slope form initially as we have a point ($B$) and will calculate the slope. These fundamental concepts form the bedrock of our approach to solving the problem. By understanding these principles, we can navigate the solution process with clarity and precision. This conceptual clarity will not only aid in solving this particular problem but also in tackling a wide array of geometry problems. Therefore, a firm grasp of these concepts is essential for anyone venturing into the world of coordinate geometry.

Step-by-Step Solution

Now, let's embark on a step-by-step solution to determine the equation of $\overrightarrow{BC}$.

Step 1: Calculate the slope of $\overrightarrow{AB}$

The first step involves calculating the slope of the line segment $\overrightarrow{AB}$. Given the coordinates of points $A(-3, -1)$ and $B(4, 4)$, we can use the slope formula:

$$m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-1)}{4 - (-3)} = \frac{5}{7}$$

Thus, the slope of $\overrightarrow{AB}$ is $\frac{5}{7}$. This calculation is crucial as it sets the stage for finding the slope of the perpendicular line, $\overrightarrow{BC}$. The slope of a line provides vital information about its orientation and steepness in the coordinate plane. By accurately calculating the slope of $\overrightarrow{AB}$, we establish a solid foundation for the subsequent steps in our solution. The formula used here, derived from the fundamental principles of coordinate geometry, is a cornerstone in analyzing linear relationships. This initial calculation not only provides a numerical value but also bridges the gap between the geometric representation and the algebraic expression of the line. The accuracy of this step is paramount, as any error here will propagate through the rest of the solution, leading to an incorrect final answer. Therefore, we emphasize the importance of meticulous calculation and a thorough understanding of the slope formula.

Step 2: Determine the slope of $\overrightarrow{BC}$

Since $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are perpendicular, the slope of $\overrightarrow{BC}$ ($m_{BC}$) is the negative reciprocal of the slope of $\overrightarrow{AB}$ ($m_{AB}$). Therefore:

$$m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5}$$

So, the slope of $\overrightarrow{BC}$ is $-\frac{7}{5}$. This step leverages the fundamental geometric principle that perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal relationship is a direct consequence of the definition of perpendicularity and the properties of slopes in the Cartesian plane. Understanding this relationship is crucial for solving problems involving right angles and orthogonal lines. The calculation here not only provides the numerical value of the slope of $\overrightarrow{BC}$ but also demonstrates the application of a core geometric concept. This slope, $-\frac{7}{5}$, indicates that the line $\overrightarrow{BC}$ descends as it moves from left to right, contrasting with the ascending line $\overrightarrow{AB}$. This visual understanding of the slopes adds another layer of comprehension to the problem. The accurate determination of this slope is essential for the subsequent formulation of the equation of the line $\overrightarrow{BC}$.

Step 3: Find the equation of $\overrightarrow{BC}$

Now that we have the slope of $\overrightarrow{BC}$ (-rac{7}{5}) and a point on the line, $B(4, 4)$, we can use the point-slope form of a linear equation:

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

Substituting the values, we get:

$$y - 4 = -\frac{7}{5}(x - 4)$$

Let's convert this to the standard form, which is $Ax + By = C$. Multiplying both sides by 5, we get:

$$5(y - 4) = -7(x - 4)$$

$$5y - 20 = -7x + 28$$

Rearranging the terms, we have:

$$7x + 5y = 48$$

Step 4: Verify the Answer

To verify our answer, let's compare our result with the given options. The equation $7x + 5y = 48$ does not directly match any of the options. However, if we multiply both sides of the equation by -1, we get $-7x - 5y = -48$, which corresponds to option C when the equation is rearranged. This process of transforming the equation into a comparable form underscores the importance of algebraic manipulation in problem-solving. The initial mismatch between our derived equation and the provided options highlights the need for careful comparison and possible rearrangement. By multiplying both sides by -1, we reveal the equivalence between our solution and option C, thus confirming the correctness of our approach. This step emphasizes that mathematical problems often have multiple equivalent solutions, and the ability to recognize these equivalences is a valuable skill. The verification process not only confirms the solution but also reinforces the understanding of algebraic principles and their application in geometry.

Final Answer

Therefore, the equation of $\overrightarrow{BC}$ is $-7x - 5y = -48$, which corresponds to option C.

Conclusion

In conclusion, we have successfully determined the equation of the line $\overrightarrow{BC}$ by systematically applying geometric principles and algebraic techniques. This problem showcased the importance of understanding the relationship between slopes of perpendicular lines and the application of the point-slope form of a linear equation. By breaking down the problem into manageable steps, we were able to navigate through the solution process with clarity and precision. The initial step involved calculating the slope of $\overrightarrow{AB}$, which served as the foundation for finding the slope of the perpendicular line, $\overrightarrow{BC}$. Leveraging the negative reciprocal relationship between perpendicular lines, we accurately determined the slope of $\overrightarrow{BC}$. Subsequently, we utilized the point-slope form of a linear equation, incorporating the calculated slope and the coordinates of point $B$, to formulate the equation of $\overrightarrow{BC}$. The final step involved transforming the equation into the standard form and comparing it with the provided options, ensuring the accuracy of our solution. This comprehensive approach not only yielded the correct answer but also reinforced the understanding of fundamental geometric and algebraic concepts. The ability to apply these principles in solving problems is a testament to the power of mathematical reasoning and its practical applications in coordinate geometry.