Equation Of Line Through Midpoint With Given Gradient A Comprehensive Solution

• Introduction
• Finding the Midpoint M
• Calculating the Gradient
• Determining the Equation of the Line
• Solution
• Conclusion

Introduction

In coordinate geometry, determining the equation of a line is a fundamental concept. This often involves using given points and gradients to form the equation in various forms such as slope-intercept form, point-slope form, or the general form. This article will delve into a specific problem: finding the equation of a line that passes through the midpoint of two given points and has a given gradient. This problem integrates the concepts of finding the midpoint of a line segment and using the point-slope form to derive the equation of a line.

The problem presented here is a classic example of how coordinate geometry combines algebraic and geometric concepts. Understanding these principles is crucial not only for academic success but also for various applications in fields like engineering, computer graphics, and physics. This article aims to provide a detailed, step-by-step solution to the problem, ensuring clarity and understanding for readers of all levels. We will break down each component, from finding the midpoint to applying the gradient, and finally formulating the equation of the line. This methodical approach will serve as a valuable guide for tackling similar problems in the future. The equation of a line through a midpoint given specific points and gradient is indeed a fascinating problem in coordinate geometry. To fully address it, we will go through several steps, each building upon the previous one to arrive at the final solution. The core concepts involved here are finding the midpoint of a line segment, understanding the gradient (or slope) of a line, and using the point-slope form to derive the equation of the line. By meticulously working through each of these steps, we can develop a comprehensive understanding and arrive at the correct equation. So let’s start by revisiting the fundamental concepts and then applying them to the specific problem at hand. This exploration will not only solve the immediate question but also enhance our understanding of coordinate geometry principles.

Finding the Midpoint M

To begin, we need to find the midpoint $M$ of the line segment connecting the points $(-1, 4)$ and $(5, 8)$. The midpoint formula is a fundamental concept in coordinate geometry, and it's derived from the idea of averaging the x-coordinates and the y-coordinates of the two points. This formula allows us to find the exact center of the line segment, which is crucial for our problem. The formula is given by:

$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points. In our case, $(x_1, y_1) = (-1, 4)$ and $(x_2, y_2) = (5, 8)$. Now, let's apply the formula step by step. We first sum the x-coordinates: $-1 + 5 = 4$. Then, we divide this sum by 2 to get the x-coordinate of the midpoint: $4 / 2 = 2$. Next, we sum the y-coordinates: $4 + 8 = 12$. Similarly, we divide this sum by 2 to get the y-coordinate of the midpoint: $12 / 2 = 6$. Therefore, the midpoint $M$ is $(2, 6)$. This is a crucial step because the line we are trying to find passes through this exact point. A slight miscalculation here would lead to a completely different line. The midpoint formula is a direct application of the concept of averaging. When we find the midpoint, we are essentially finding the average position between two points in a two-dimensional plane. This concept is not only useful in coordinate geometry but also has applications in other areas of mathematics and physics, such as finding the center of mass of a system. Understanding the intuition behind the formula helps in remembering it and applying it correctly. So, by averaging the x-coordinates and the y-coordinates separately, we accurately locate the point that is equidistant from both endpoints of the line segment. The midpoint, with coordinates (2, 6), serves as a crucial anchor for our next steps in determining the equation of the line. Now that we have the coordinates of the midpoint, we can proceed to incorporating the given gradient to formulate the line's equation.

Calculating the Gradient

In this problem, the gradient of the line is given as m = -rac{1}{2}. The gradient, often denoted as 'm', is a measure of the steepness and direction of a line. 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. In other words, it tells us how much the line rises (or falls) for every unit increase in the horizontal direction. A positive gradient indicates that the line slopes upwards from left to right, while a negative gradient indicates that it slopes downwards. A gradient of zero means the line is horizontal, and an undefined gradient (infinite) means the line is vertical. In our specific case, the gradient m = -rac{1}{2} tells us that the line slopes downwards. For every 2 units we move horizontally, the line drops 1 unit vertically. This negative value is crucial for the orientation of the line we are trying to define. The gradient is a fundamental concept in linear equations and coordinate geometry. It is not just a number; it represents a critical property of the line. Understanding the gradient is essential for interpreting the behavior of linear functions and for various applications in fields like physics, engineering, and economics. For instance, in physics, the gradient can represent the slope of a velocity-time graph, giving us the acceleration. In economics, it can represent the rate of change of cost or revenue. The given gradient of -rac{1}{2} simplifies our task significantly because we don't have to calculate it from two points on the line. Instead, we can directly use it in the point-slope form of the line equation. This makes the problem more straightforward and allows us to focus on the remaining steps of formulating the equation. Now, with the midpoint $M(2, 6)$ and the gradient m = -rac{1}{2} in hand, we are well-equipped to determine the equation of the line that meets these conditions. The next step involves using this information to apply the point-slope form, which is perfectly suited for this scenario.

Determining the Equation of the Line

To determine the equation of the line, we will use the point-slope form. The point-slope form is a versatile way to represent the equation of a line when we know one point on the line and its gradient. This form is particularly useful in situations like ours, where we have the midpoint and the gradient. 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 gradient. In our case, we have the midpoint $M(2, 6)$ as our point $(x_1, y_1)$, and the gradient m = -rac{1}{2}. Now, we substitute these values into the point-slope form: y - 6 = -rac{1}{2}(x - 2). This equation represents the line we are looking for, but it's not in the standard form that's usually presented in answer choices (like $Ax + By + C = 0$). Therefore, the next step is to rearrange this equation into a more familiar form. To do this, we first multiply both sides of the equation by 2 to eliminate the fraction: $2(y - 6) = -1(x - 2)$. This simplifies to: $2y - 12 = -x + 2$. Now, we want to get all the terms on one side of the equation to set it equal to zero. We add $x$ to both sides and subtract 2 from both sides: $x + 2y - 12 - 2 = 0$. Finally, we simplify to get: $x + 2y - 14 = 0$. This is the equation of the line in the general form $Ax + By + C = 0$. The point-slope form is powerful because it directly incorporates the geometric information of a point and the gradient into an algebraic equation. It's a fundamental tool in coordinate geometry and allows us to easily derive the equation of a line under various conditions. Understanding and applying the point-slope form is essential for solving a wide range of problems in mathematics and related fields. By rearranging the point-slope form into the general form, we have made the equation more accessible and comparable to standard forms. The general form is often preferred because it clearly shows the coefficients of x and y, which can be useful for further analysis or comparisons with other lines. Now that we have the equation in the general form, we can confidently state our solution and compare it with the given options.

Solution

After following the steps to find the midpoint and using the given gradient, we arrived at the equation of the line as:

$$x + 2y - 14 = 0$$

Comparing this equation with the given options:

A. $x + 2y - 10 = 0$ B. $x + 2y - 14 = 0$ C. $2x + y - 10 = 0$ D. $2x - y - 14 = 0$

It is clear that option B, $x + 2y - 14 = 0$, matches our derived equation. Therefore, option B is the correct answer. This confirms that our methodical approach and calculations were accurate. The solution process highlights the importance of careful and precise steps in solving mathematical problems. Each step, from finding the midpoint to applying the point-slope form and rearranging the equation, needs to be executed correctly to arrive at the right answer. A small error in any step can lead to a different result, so it's crucial to double-check each calculation and ensure that the logic is sound. Moreover, understanding the underlying concepts, such as the midpoint formula and the point-slope form, is essential for tackling such problems effectively. Rote memorization of formulas is not sufficient; a deep understanding of the principles allows for flexible application and problem-solving. The final step of comparing our solution with the given options is also important. It's a way to verify our answer and ensure that we have not made any algebraic errors in the rearrangement of the equation. This step provides an additional layer of confidence in our solution. In conclusion, the problem is not just about arriving at the correct answer; it's also about the process and the understanding gained along the way. The systematic approach we followed can be applied to a wide range of coordinate geometry problems, making it a valuable skill for mathematical problem-solving.

Conclusion

In conclusion, by systematically applying the midpoint formula and the point-slope form, we successfully determined the equation of the line passing through the midpoint $M$ of the points $(-1, 4)$ and $(5, 8)$ with a gradient of -rac{1}{2}. The equation of the line is:

$$x + 2y - 14 = 0$$

This corresponds to option B in the given choices. This problem underscores the interconnectedness of geometric and algebraic concepts in coordinate geometry. The ability to seamlessly transition between geometric ideas and their algebraic representations is crucial for solving such problems. The solution process not only provides the answer but also reinforces the understanding of fundamental principles. The methodical approach demonstrated here can be applied to various problems in coordinate geometry, highlighting the importance of a step-by-step problem-solving strategy. Each step, from finding the midpoint to using the point-slope form and rearranging the equation, played a critical role in reaching the final solution. Furthermore, the ability to interpret and apply mathematical formulas and concepts in a logical sequence is a key skill that extends beyond this specific problem. It is a valuable asset for tackling more complex mathematical challenges. The midpoint formula and the point-slope form are powerful tools in coordinate geometry, and mastering their application is essential for success in this area. These tools, combined with a clear understanding of gradients and linear equations, enable us to solve a wide range of problems involving lines and points in the Cartesian plane. In summary, this exercise not only solves a specific problem but also enhances our overall understanding of coordinate geometry and problem-solving strategies. It reinforces the importance of precision, methodical thinking, and the ability to connect geometric intuition with algebraic manipulation. By mastering these skills, students and enthusiasts can confidently approach similar problems and deepen their appreciation for the beauty and utility of mathematics. The final answer, $x + 2y - 14 = 0$, is a testament to the effectiveness of a structured approach and a solid grasp of fundamental concepts.