Equation Of New Path Perpendicular To Existing Path And Passing Through F(4, 14)

In this article, we embark on a fascinating mathematical journey to determine the equation of a new path leading to a fountain. This path, designed to enhance accessibility, will be perpendicular to an existing path and will traverse through a specific point, F(4, 14). Our exploration will delve into the fundamental concepts of coordinate geometry, including the determination of slopes, the application of perpendicularity conditions, and the derivation of linear equations. This comprehensive analysis will not only provide a solution to the problem at hand but also offer valuable insights into the practical applications of mathematical principles in real-world scenarios.

To embark on our quest to define the new path, we must first possess a clear understanding of the existing path. Unfortunately, the details of this existing path remain elusive at this juncture. We lack crucial information such as its equation, slope, or even a graphical representation. To surmount this obstacle, we must assume that the existing path is a straight line. This assumption, grounded in the principles of Euclidean geometry, allows us to leverage the well-established properties of linear equations. A straight line can be uniquely defined by two key parameters: its slope and its y-intercept. The slope, a measure of the line's steepness, dictates the rate at which the line rises or falls. The y-intercept, on the other hand, marks the point where the line intersects the vertical axis. To proceed further, we will assume that the equation of the existing path is given by y = mx + c, where m represents the slope and c represents the y-intercept. Without specific values for m and c, we will keep these variables symbolic, allowing our solution to remain general and adaptable to various scenarios. The subsequent steps will involve determining the slope of the new path, which is intricately linked to the slope of the existing path, and then constructing the equation that satisfies the given conditions. This methodical approach will ensure that our solution is both accurate and insightful, providing a comprehensive understanding of the geometric relationships involved. The journey ahead promises to be an engaging exploration of mathematical concepts, culminating in a precise definition of the new path to the fountain.

The crux of our problem lies in determining the equation of a new path that is not only accessible but also perpendicular to the existing path. This perpendicularity condition is a cornerstone of our solution, dictating a specific relationship between the slopes of the two paths. In the realm of coordinate geometry, two lines are deemed perpendicular if and only if the product of their slopes equals -1. This elegant mathematical principle provides us with a direct means to calculate the slope of the new path, given the slope of the existing path. Let us denote the slope of the existing path as m. As we established earlier, the equation of the existing path is assumed to be y = mx + c. Now, let us represent the slope of the new path as m'. According to the perpendicularity condition, the product of m and m' must be -1. This can be expressed mathematically as m * m' = -1. From this equation, we can readily deduce the slope of the new path: m' = -1/m. This formula is a pivotal result, allowing us to compute the slope of the new path as soon as we know the slope of the existing path. The negative reciprocal relationship between the slopes of perpendicular lines is a fundamental concept in geometry, and its application here is crucial for defining the direction of the new path. By understanding this relationship, we can ensure that the new path intersects the existing path at a right angle, fulfilling the design requirement of perpendicularity. The next step in our journey involves utilizing this newly acquired knowledge of the slope, along with the given point F(4, 14), to construct the equation of the new path. This equation will provide a complete mathematical description of the path, allowing for precise navigation and construction.

With the slope of the new path firmly in our grasp, we now turn our attention to the task of constructing its equation. The point-slope form of a linear equation emerges as the ideal tool for this purpose. This form, a cornerstone of coordinate geometry, elegantly captures the essence of a line's equation using its slope and a single point that lies on the line. The point-slope form is expressed as y - y₁ = m'(x - x₁), where (x₁, y₁) represents the coordinates of a known point on the line, and m' denotes the slope of the line. In our specific scenario, we have a wealth of information at our disposal. We know that the new path must pass through the point F(4, 14), which provides us with the coordinates x₁ = 4 and y₁ = 14. Furthermore, we have already determined the slope of the new path, m', as the negative reciprocal of the slope of the existing path, expressed as m' = -1/m. By substituting these values into the point-slope form, we obtain the equation of the new path: y - 14 = (-1/m)(x - 4). This equation, while seemingly simple, encapsulates the geometric relationship between the new path and the existing path, ensuring that they intersect perpendicularly and that the new path passes through the designated point F(4, 14). To further refine this equation, we can rearrange it into the slope-intercept form, y = mx + c, which provides a clearer visualization of the line's slope and y-intercept. This rearrangement involves algebraic manipulation, distributing the slope and isolating y on one side of the equation. The resulting equation will be a precise mathematical representation of the new path, allowing for accurate mapping and construction. The journey from the point-slope form to the slope-intercept form is a testament to the power of algebraic manipulation in unraveling the intricacies of geometric relationships.

To gain a clearer understanding of the new path's characteristics, we can transform the equation from point-slope form to slope-intercept form. The slope-intercept form, y = m'x + b, explicitly reveals the slope (m') and y-intercept (b) of the line, providing valuable insights into its orientation and position on the coordinate plane. Starting with the equation derived in the previous section, y - 14 = (-1/m)(x - 4), we embark on a series of algebraic manipulations to isolate y on one side. First, we distribute the term (-1/m) across the parentheses, resulting in y - 14 = (-1/m)x + 4/m. Next, we add 14 to both sides of the equation to eliminate the constant term on the left side, yielding y = (-1/m)x + 4/m + 14. This equation is now in slope-intercept form, where the slope is (-1/m), as we previously established, and the y-intercept is 4/m + 14. The y-intercept represents the point where the new path intersects the y-axis. By analyzing this value, we can determine the vertical position of the path relative to the origin. The slope-intercept form provides a concise and readily interpretable representation of the new path's equation, allowing us to visualize its trajectory and understand its relationship to the existing path. This form is particularly useful for graphing the line and for making comparisons with other linear equations. The transition from point-slope form to slope-intercept form exemplifies the versatility of algebraic techniques in transforming mathematical expressions into more informative representations. The final equation, y = (-1/m)x + 4/m + 14, encapsulates the complete mathematical description of the new path, paving the way for its physical implementation.

To solidify our understanding of the concepts and equations we have explored, let's consider a concrete example. Suppose the existing path has a slope of m = 2. This implies that for every unit increase in the x-coordinate, the y-coordinate increases by 2 units. Now, let's apply our derived formulas to determine the equation of the new path, which, as we know, must be perpendicular to the existing path and pass through the point F(4, 14). First, we calculate the slope of the new path, m', using the formula m' = -1/m. Substituting m = 2, we get m' = -1/2. This indicates that the new path has a negative slope, meaning it slopes downwards as we move from left to right. The magnitude of the slope, 1/2, signifies that for every two-unit increase in the x-coordinate, the y-coordinate decreases by one unit. Next, we substitute the values of m', x₁, and y₁ into the point-slope form of the equation: y - 14 = (-1/2)(x - 4). To simplify this equation into slope-intercept form, we distribute the term (-1/2) across the parentheses, resulting in y - 14 = (-1/2)x + 2. Then, we add 14 to both sides to isolate y, obtaining y = (-1/2)x + 16. This equation, y = (-1/2)x + 16, represents the new path in slope-intercept form. We can readily see that the slope is -1/2, as expected, and the y-intercept is 16. This means that the new path intersects the y-axis at the point (0, 16). By graphing this equation, we can visually confirm that it is indeed perpendicular to a line with a slope of 2 and that it passes through the point F(4, 14). This illustrative example demonstrates the practical application of our derived formulas and provides a tangible understanding of the geometric relationships involved. The ability to translate abstract mathematical concepts into concrete scenarios is a hallmark of effective problem-solving.

In conclusion, we have successfully navigated the mathematical landscape to derive the equation of the new path, a path designed to enhance access to the fountain while adhering to the critical requirement of perpendicularity. Our journey commenced with an exploration of the existing path, where we established the foundational assumption of linearity. We then delved into the concept of perpendicularity, leveraging the principle that the product of the slopes of perpendicular lines is -1. This principle paved the way for the calculation of the new path's slope, a pivotal step in our solution. Subsequently, we harnessed the power of the point-slope form to construct the initial equation of the new path, incorporating the given point F(4, 14) and the calculated slope. To gain a clearer perspective on the path's characteristics, we transformed the equation into slope-intercept form, revealing the slope and y-intercept with clarity. An illustrative example served to solidify our understanding, demonstrating the practical application of our derived formulas in a concrete scenario. The culmination of our efforts is the equation y = (-1/m)x + 4/m + 14, or, in the specific case where the existing path has a slope of 2, the equation y = (-1/2)x + 16. These equations represent the new path, a testament to the power of mathematical reasoning and problem-solving. The path, now well-defined, stands ready to be implemented, providing enhanced access to the fountain while adhering to the geometric constraints. This endeavor underscores the vital role of mathematics in shaping our physical world, transforming abstract concepts into tangible realities.