Finding The Equation Of A Line Forming A Right Angle A Comprehensive Guide

In the realm of geometry, understanding the relationships between lines, angles, and coordinates is fundamental. This article delves into a specific problem involving lines forming a right angle and aims to determine the equation of a line given the coordinates of two points on it. Let's explore the concepts of right angles, linear equations, and how to apply them to solve this problem.

Decoding the Problem: Lines, Right Angles, and Coordinates

The problem states that lines $\overleftrightarrow{A B}$ and $\overleftrightarrow{B C}$ intersect at point $B$ and form a right angle. This crucial piece of information tells us that the two lines are perpendicular to each other. Additionally, we are given the coordinates of points $A$ and $B$ as $(14, -1)$ and $(2, 1)$, respectively. Our objective is to find the equation of the line $\overleftrightarrow{A B}$. To achieve this, we will utilize our knowledge of linear equations and the properties of perpendicular lines. Understanding the relationship between lines and angles is crucial in solving geometric problems, as it allows us to leverage the properties of specific angles like right angles to derive further information about the lines themselves. In this case, the fact that $\overleftrightarrow{A B}$ and $\overleftrightarrow{B C}$ form a right angle tells us that they are perpendicular, a key concept we'll use later. Furthermore, knowing the coordinates of points A and B allows us to calculate the slope of the line $\overleftrightarrow{A B}$, which is a critical component in determining its equation. The interplay between these geometric and algebraic concepts is what makes this problem both interesting and solvable.

The Slope-Intercept Form: A Foundation for Linear Equations

Before we dive into the solution, let's revisit the slope-intercept form of a linear equation, which is expressed as $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the y-intercept (the point where the line crosses the y-axis). The slope, $m$, quantifies the steepness of the line and is calculated as the change in $y$ divided by the change in $x$ between any two points on the line. The y-intercept, $b$, provides the vertical position of the line. This form is particularly useful because it directly reveals two key characteristics of the line: its rate of change (slope) and its starting point on the y-axis (y-intercept). To effectively use the slope-intercept form, it’s essential to understand how to calculate the slope given two points and how to substitute a point’s coordinates into the equation to solve for the y-intercept. Understanding the slope-intercept form is not just about memorizing the equation; it's about grasping the geometric meaning behind the symbols. The slope, m, tells us how much the y-value changes for every unit change in the x-value, which visually translates to the steepness and direction of the line. The y-intercept, b, anchors the line to a specific point on the y-axis, defining its vertical position. Mastering this form allows us to move seamlessly between the algebraic representation of a line and its graphical depiction, making it a fundamental tool in coordinate geometry. This understanding is crucial for solving a wide range of problems, from finding the equation of a line given two points to analyzing the behavior of linear functions.

Calculating the Slope: The Rise Over Run

To determine the equation of $\overleftrightarrow{A B}$, our first step is to calculate its slope. Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope, $m$, is calculated using the formula: $m = (y_2 - y_1) / (x_2 - x_1)$. Applying this to points $A(14, -1)$ and $B(2, 1)$, we have:

$m = (1 - (-1)) / (2 - 14) = 2 / (-12) = -1/6$

Therefore, the slope of $\overleftrightarrow{A B}$ is $-1/6$. The process of calculating the slope is a fundamental step in finding the equation of a line. It's not just about plugging numbers into a formula; it's about understanding the concept of "rise over run." The numerator, $y_2 - y_1$, represents the vertical change (rise) between the two points, while the denominator, $x_2 - x_1$, represents the horizontal change (run). The ratio of these changes gives us the slope, which quantifies the line's steepness and direction. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line. In our case, the negative slope of -1/6 tells us that the line $\overleftrightarrow{A B}$ slopes downward from left to right. This visual interpretation of the slope helps us to check the reasonableness of our calculation and to better understand the line's behavior on the coordinate plane. This calculated slope will be crucial in determining the equation of the line in the next steps.

Finding the Equation: Using the Slope-Intercept Form

Now that we have the slope, $m = -1/6$, we can substitute this into the slope-intercept form: $y = mx + b$, giving us $y = (-1/6)x + b$. To find the y-intercept, $b$, we can substitute the coordinates of either point $A$ or $B$ into this equation. Let's use point $B(2, 1)$:

$1 = (-1/6)(2) + b$

$1 = -1/3 + b$

$b = 1 + 1/3 = 4/3$

Thus, the y-intercept is $4/3$. Now we can write the complete equation of $\overleftrightarrow{A B}$ as:

$y = (-1/6)x + 4/3$

The final step of finding the equation involves putting all the pieces together. We've calculated the slope and the y-intercept, and now we can substitute these values into the slope-intercept form to obtain the equation of the line. This process is not just about algebraic manipulation; it's about synthesizing our understanding of slope, y-intercept, and the general form of a linear equation. The equation $y = (-1/6)x + 4/3$ encapsulates all the information we've gathered about the line $\overleftrightarrow{A B}$. It tells us the line's steepness, direction, and where it crosses the y-axis. We can use this equation to find any point on the line, to graph the line, or to analyze its relationship with other lines. The power of the equation lies in its ability to represent a geometric object, a line, in a concise and easily manipulated algebraic form. This equation is the culmination of our problem-solving process and provides a complete description of the line $\overleftrightarrow{A B}$.

The Solution: Unveiling the Equation

Therefore, the equation of $\overleftrightarrow{A B}$ is $y = (-1/6)x + 4/3$. This equation represents a line with a slope of $-1/6$ and a y-intercept of $4/3$. This solution demonstrates how we can combine geometric concepts and algebraic techniques to solve problems involving lines and coordinates. Understanding the solution in the context of the problem is just as important as the algebraic steps taken to arrive at it. The equation $y = (-1/6)x + 4/3$ is not just a string of symbols; it's a concise representation of the line $\overleftrightarrow{A B}$ on the coordinate plane. The negative slope tells us that the line slopes downward from left to right, and the y-intercept of 4/3 tells us that the line crosses the y-axis at the point (0, 4/3). We can visualize this line, and it should align with our understanding of the problem: it passes through points A(14, -1) and B(2, 1), and it forms a right angle with line $\overleftrightarrow{B C}$. This connection between the algebraic equation and the geometric representation is a key aspect of coordinate geometry. By verifying that our solution makes sense in the context of the original problem, we can build confidence in our understanding and problem-solving skills.

Importance of Understanding Linear Equations in Geometry

This problem highlights the importance of understanding linear equations in geometry. By mastering concepts like slope, y-intercept, and the slope-intercept form, we can effectively analyze and solve geometric problems involving lines and coordinates. Furthermore, the ability to connect geometric properties, such as right angles, with algebraic representations allows us to tackle a wider range of problems. The importance of understanding linear equations extends far beyond the specific problem we've solved here. Linear equations are the building blocks of many mathematical and scientific models. They are used to describe relationships between variables that change at a constant rate, and they appear in a wide range of applications, from physics and engineering to economics and computer science. In geometry, linear equations are essential for describing lines, planes, and other geometric objects. They allow us to translate geometric concepts into algebraic expressions, which can then be manipulated and analyzed using powerful algebraic techniques. This ability to bridge the gap between geometry and algebra is a cornerstone of mathematical thinking, and it opens the door to solving complex problems in a variety of fields. By developing a strong understanding of linear equations, we equip ourselves with a valuable tool for problem-solving and critical thinking.

Conclusion: A Blend of Geometry and Algebra

In conclusion, this problem demonstrates the power of combining geometric principles with algebraic techniques to find the equation of a line. By understanding the concepts of slope, y-intercept, and the slope-intercept form, and by applying these concepts to the given information about points and angles, we successfully determined the equation of $\overleftrightarrow{A B}$. This exercise reinforces the interconnectedness of geometry and algebra and highlights the importance of a strong foundation in both areas for effective problem-solving. The conclusion of this problem reinforces the central theme of the article: the powerful synergy between geometry and algebra. We began with a geometric problem involving lines and angles, and we used algebraic techniques to arrive at a solution. This process highlights the importance of a strong foundation in both areas of mathematics. The ability to translate geometric concepts into algebraic expressions, and vice versa, is a crucial skill for problem-solving. It allows us to leverage the strengths of both disciplines to tackle complex challenges. By mastering these fundamental concepts and techniques, we not only enhance our mathematical abilities but also develop our critical thinking skills and our ability to approach problems from multiple perspectives. This problem serves as a reminder that mathematics is not a collection of isolated topics but a unified body of knowledge, where different areas are interconnected and mutually supportive. By embracing this interconnectedness, we can unlock the full potential of our mathematical understanding.