Finding The X-Intercept Of A Perpendicular Line A Step-by-Step Guide

In the realm of coordinate geometry, understanding the relationships between lines is fundamental. One such relationship, perpendicularity, plays a crucial role in various geometric problems. When two lines are perpendicular, they intersect at a right angle, which translates to a specific relationship between their slopes. This article delves into a problem involving perpendicular lines, focusing on how to determine the x-intercept of a line given its perpendicularity to another line and a point it passes through. This problem serves as an excellent exercise in applying key concepts of coordinate geometry, such as the slope-intercept form of a line, the relationship between slopes of perpendicular lines, and the method for finding the equation of a line. The specific problem we will address involves a line, denoted as $\overleftrightarrow{CD}$, that is perpendicular to another line, $\overleftrightarrow{AB}$. We are given the coordinates of points A and B, as well as a point C that lies on $\overleftrightarrow{CD}$. The objective is to find the x-intercept of $\overleftrightarrow{CD}$. This requires a step-by-step approach, starting with determining the slope of $\overleftrightarrow{AB}$, then finding the slope of $\overleftrightarrow{CD}$, and finally deriving the equation of $\overleftrightarrow{CD}$ to identify its x-intercept. The x-intercept is the point where the line crosses the x-axis, meaning the y-coordinate at this point is zero. By setting the y-value in the line's equation to zero, we can solve for the x-coordinate, which gives us the x-intercept. This article aims to provide a clear and detailed explanation of the solution process, making it accessible to students and enthusiasts of mathematics. Understanding these concepts is essential for tackling more complex problems in geometry and related fields. By mastering the techniques involved in this problem, readers will gain a solid foundation in coordinate geometry and enhance their problem-solving skills. The following sections will break down the problem into manageable steps, providing explanations and calculations along the way. Let's embark on this mathematical journey to unravel the solution and gain a deeper understanding of perpendicular lines and their properties.

Understanding the Problem: Perpendicular Lines and X-Intercepts

Before diving into the solution, let's clarify the core concepts involved in this problem. The problem states that line $\overleftrightarrow{CD}$ is perpendicular to line $\overleftrightarrow{AB}$. In coordinate geometry, two lines are perpendicular if and only if the product of their slopes is -1. This is a crucial piece of information that allows us to relate the slopes of the two lines. If we denote the slope of $\overleftrightarrow{AB}$ as $m_{AB}$ and the slope of $\overleftrightarrow{CD}$ as $m_{CD}$, then the condition for perpendicularity can be written as: $m_AB} \cdot m_{CD} = -1$ This relationship is the cornerstone of our approach. We are given the coordinates of points A and B, which lie on $\overleftrightarrow{AB}$. This allows us to calculate the slope of $\overleftrightarrow{AB}$ using the slope formula $m = \frac{y_2 - y_1x_2 - x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of two points on the line. Once we have the slope of $\overleftrightarrow{AB}$, we can use the perpendicularity condition to find the slope of $\overleftrightarrow{CD}$. Another key element of the problem is the x-intercept. The x-intercept of a line is the point where the line intersects the x-axis. At this point, the y-coordinate is always zero. Therefore, to find the x-intercept, we need to find the x-coordinate when y = 0. This typically involves finding the equation of the line in slope-intercept form (y = mx + b) or point-slope form (y - y1 = m(x - x1)) and then substituting y = 0 and solving for x. The problem also provides us with a point C that lies on $\overleftrightarrow{CD}$. This point, along with the slope of $\overleftrightarrow{CD}$, will enable us to determine the equation of $\overleftrightarrow{CD}$. We can use the point-slope form of a line, which is particularly useful when we have a point and the 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. By substituting the coordinates of point C and the slope of $\overleftrightarrow{CD$ into this equation, we can find the equation of $\overleftrightarrow{CD}$. Then, setting y = 0 will allow us to solve for the x-intercept. In summary, this problem requires us to combine our understanding of slopes, perpendicular lines, and the equation of a line to find a specific point, the x-intercept. By breaking the problem down into smaller steps, we can systematically arrive at the solution. The next section will detail the step-by-step solution, demonstrating how to apply these concepts to find the x-intercept of $\overleftrightarrow{CD}$.

Step-by-Step Solution: Finding the X-Intercept

Now, let's break down the solution into manageable steps to find the x-intercept of line $\overleftrightarrow{CD}$.

Step 1: Calculate the Slope of $\overleftrightarrow{AB}$

We are given the coordinates of points A and B as (-10, -3) and (7, 14), respectively. To find the slope ($m_{AB}$) of $\overleftrightarrow{AB}$, we use the slope formula:

$$m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}$$

Substituting the coordinates of A and B, we get:

$$m_{AB} = \frac{14 - (-3)}{7 - (-10)} = \frac{17}{17} = 1$$

Thus, the slope of $\overleftrightarrow{AB}$ is 1. This means that for every one unit we move to the right along the line, we move one unit up. This positive slope indicates that the line is increasing from left to right. Understanding the slope is crucial because it dictates the direction and steepness of the line, which is essential for further calculations involving perpendicular lines and their properties. The accurate calculation of this slope is the first key step in solving the problem, as it lays the foundation for determining the slope of the perpendicular line $\overleftrightarrow{CD}$ in the subsequent steps. The simplicity of this slope (1) can sometimes be deceiving, but it's a straightforward value that allows for easy manipulation in the following calculations. It's also important to note the significance of the positive slope, as it provides a visual understanding of the line's orientation in the coordinate plane. Having determined the slope of $\overleftrightarrow{AB}$, we are now well-equipped to move on to the next step, which involves finding the slope of the line perpendicular to it.

Step 2: Determine the Slope of $\overleftrightarrow{CD}$

Since $\overleftrightarrow{CD}$ is perpendicular to $\overleftrightarrow{AB}$, the product of their slopes is -1. We know the slope of $\overleftrightarrow{AB}$ ($m_{AB}$) is 1. Let the slope of $\overleftrightarrow{CD}$ be $m_{CD}$. Then:

$$m_{AB} \cdot m_{CD} = -1$$

Substituting $m_{AB} = 1$, we get:

$$1 \cdot m_{CD} = -1$$

$$m_{CD} = -1$$

So, the slope of $\overleftrightarrow{CD}$ is -1. This negative slope indicates that the line is decreasing from left to right, meaning that as we move along the line in the positive x-direction, the y-values decrease. The fact that the slope is -1 also signifies that for every one unit we move to the right, we move one unit down. This is the exact opposite of the slope of $\overleftrightarrow{AB}$, which is expected given that the lines are perpendicular. The relationship between the slopes of perpendicular lines is a fundamental concept in coordinate geometry, and this step demonstrates the practical application of that concept. Understanding this relationship is essential not only for solving this particular problem but also for tackling a wide range of geometric challenges. By accurately determining the slope of $\overleftrightarrow{CD}$, we have set the stage for finding the equation of the line, which will be the next crucial step in locating the x-intercept. The slope of -1 for $\overleftrightarrow{CD}$ is a straightforward value that simplifies the subsequent calculations, making the process of finding the equation of the line more manageable.

Step 3: Find the Equation of $\overleftrightarrow{CD}$

We know that $\overleftrightarrow{CD}$ passes through point C(5, 12) and has a slope ($m_{CD}$) of -1. We can use the point-slope form of a line to find the equation:

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

Substituting the coordinates of C (5, 12) and the slope -1, we get:

$$y - 12 = -1(x - 5)$$

Simplifying the equation:

$$y - 12 = -x + 5$$

$$y = -x + 17$$

Thus, the equation of $\overleftrightarrow{CD}$ is $y = -x + 17$. This equation represents the line in slope-intercept form, which is a convenient form for identifying the slope and y-intercept of the line. The slope-intercept form is given by y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is -1, as we calculated earlier, and the y-intercept is 17. The y-intercept is the point where the line crosses the y-axis, and it occurs when x = 0. Having the equation of the line in this form is incredibly useful for various geometric and algebraic manipulations. It allows us to easily graph the line, find other points on the line, and, most importantly for this problem, find the x-intercept. The process of finding the equation of the line involves careful substitution and simplification, and this step demonstrates the importance of algebraic skills in solving geometric problems. With the equation of $\overleftrightarrow{CD}$ now determined, we are just one step away from finding the x-intercept, which is the ultimate goal of this problem.

Step 4: Determine the X-Intercept of $\overleftrightarrow{CD}$

The x-intercept is the point where the line crosses the x-axis, which means the y-coordinate at this point is 0. To find the x-intercept, we set y = 0 in the equation of $\overleftrightarrow{CD}$:

$$y = -x + 17$$

$$0 = -x + 17$$

Solving for x:

$$x = 17$$

Therefore, the x-intercept of $\overleftrightarrow{CD}$ is 17. This means that the line $\overleftrightarrow{CD}$ intersects the x-axis at the point (17, 0). Finding the x-intercept is a crucial step in understanding the behavior of a line and its relationship to the coordinate axes. It represents a specific point where the line crosses the x-axis, providing valuable information about the line's position and orientation in the coordinate plane. The process of setting y = 0 and solving for x is a standard technique in algebra and coordinate geometry, and it is widely used in various applications. The result, x = 17, is a numerical value that represents the x-coordinate of the x-intercept. It is the final answer to the problem, and it encapsulates the culmination of all the previous steps. The clarity and precision of this final result highlight the importance of each step in the solution process, from calculating the slope of $\overleftrightarrow{AB}$ to finding the equation of $\overleftrightarrow{CD}$. The x-intercept, 17, provides a concrete point that can be easily visualized on a graph, further solidifying the understanding of the problem and its solution.

Conclusion

In conclusion, by systematically applying the principles of coordinate geometry, we have successfully determined the x-intercept of $\overleftrightarrow{CD}$ to be 17. This problem demonstrated the importance of understanding the relationship between slopes of perpendicular lines and the application of the point-slope form to find the equation of a line. The step-by-step approach allowed us to break down the problem into smaller, manageable parts, making the solution process clear and accessible. This exercise not only reinforces our understanding of key concepts but also enhances our problem-solving skills in mathematics. The ability to work with perpendicular lines, slopes, and intercepts is fundamental to many areas of mathematics and its applications, making this a valuable learning experience. By mastering these concepts, we are better equipped to tackle more complex problems in geometry and related fields. The solution process involved several key steps, each building upon the previous one. First, we calculated the slope of $\overleftrightarrow{AB}$ using the coordinates of points A and B. This provided a crucial piece of information for determining the slope of the perpendicular line, $\overleftrightarrow{CD}$. Next, we utilized the relationship between the slopes of perpendicular lines to find the slope of $\overleftrightarrow{CD}$. This step highlighted the importance of understanding geometric principles and their algebraic representations. Then, we used the point-slope form of a line to derive the equation of $\overleftrightarrow{CD}$. This step demonstrated the power of algebraic techniques in solving geometric problems. Finally, we set y = 0 in the equation of $\overleftrightarrow{CD}$ to find the x-intercept. This step showcased the practical application of algebraic manipulation to find specific points on a line. Throughout the solution process, we emphasized clarity, precision, and a step-by-step approach. This is crucial for effective problem-solving in mathematics, as it allows us to avoid errors and gain a deeper understanding of the underlying concepts. The x-intercept, 17, is the final answer to the problem, and it represents a specific point on the x-axis where $\overleftrightarrow{CD}$ intersects. This result not only provides a numerical solution but also enhances our geometric intuition and understanding of lines and their properties. In summary, this problem serves as an excellent example of how coordinate geometry combines algebraic and geometric concepts to solve real-world problems. By mastering these concepts and techniques, we can confidently approach a wide range of mathematical challenges.