Finding A Perpendicular Line's Equation: A Step-by-Step Guide

Hey guys! Let's dive into a geometry problem that's super interesting and useful: finding the equation of a straight line. We'll be working with coordinates, perpendicular lines, and ratios. Sounds fun, right? Specifically, we're given the coordinates of points A and C, and a line segment AC. A straight line is perpendicular to AC and intersects it at point B, with a specific ratio of AB to AC. Our mission? Find the equation of this perpendicular line. It might seem daunting at first, but trust me, we'll break it down into easy-to-follow steps. We'll start with the basics, like understanding what a perpendicular line is, then move on to calculating slopes and using the point-slope form. So, grab your pens and paper, and let's get started on this mathematical adventure! This problem is a great example of how geometry and algebra come together, and it's applicable in various real-world scenarios, from construction to computer graphics. Understanding this concept can sharpen your problem-solving skills and give you a deeper appreciation for the beauty of mathematics. We'll make sure to explain everything clearly, so even if you're new to this, you'll be able to follow along. By the end, you'll not only solve the problem, but also gain a solid understanding of the underlying principles. Ready to unlock the secrets of this geometry puzzle? Let's do it!

Understanding the Problem and Key Concepts

Alright, before we jump into calculations, let's make sure we're all on the same page. The core of this problem revolves around a few key concepts: coordinate geometry, perpendicular lines, and ratios. Let's break these down. First off, coordinate geometry deals with representing geometric shapes and figures using a coordinate system (usually the familiar x-y plane). Points are defined by their x and y coordinates, allowing us to describe lines, segments, and other shapes algebraically. Second, perpendicular lines are lines that intersect at a right angle (90 degrees). A crucial property of perpendicular lines is that their slopes are negative reciprocals of each other. This means if one line has a slope of 'm', the perpendicular line has a slope of '-1/m'. Finally, we have ratios, which express the relative sizes of two or more quantities. In this problem, the ratio AB:AC tells us how the point B divides the line segment AC. This ratio is key to determining the location of point B. By understanding these concepts, we have the foundation we need to solve the problem. Now, let's get into the specifics of what we're given. We know the coordinates of points A and C (-6, 3) and (4, 8) respectively. We also know that a line is perpendicular to AC and intersects it at point B, where the ratio of AB to AC is 2:5. This means that point B divides the segment AC in this ratio. Our ultimate goal is to find the equation of the line that passes through B and is perpendicular to AC. To find the equation, we'll need two things: a point on the line (which will be point B) and the slope of the line (which we'll derive from the fact that it is perpendicular to AC). So, keep these concepts in mind, and let's get started on the first step: finding the coordinates of point B!

Step-by-Step Solution: Finding the Equation

Now, let's roll up our sleeves and get into the actual solution. The main goal here is to find the equation of the straight line. This involves a few key steps: First, we need to find the coordinates of point B. We can do this using the section formula, which helps us find the coordinates of a point that divides a line segment in a given ratio. Second, we need to find the slope of the line segment AC. We can find the slope using the coordinates of A and C. Third, since we need the equation of a line perpendicular to AC, we will calculate the negative reciprocal of the slope we just found. This gives us the slope of the perpendicular line. Finally, once we have the slope of the perpendicular line and a point on that line (point B), we'll use the point-slope form to find the equation of the line. So, let's dive into finding those coordinates of point B. The section formula is our friend here. Given the ratio AB:AC = 2:5, we can deduce that the ratio is 2:(5-2) or 2:3. Let's name the coordinates of B as (x, y). The section formula states that: x = (mx2 + nx1)/(m+n) and y = (my2 + ny1)/(m+n). Here, (x1, y1) is the coordinate of A (-6,3) and (x2, y2) is the coordinate of C (4,8), and the ratio is m:n = 2:3. So let's calculate: x = (24 + 3(-6))/(2+3) = (8 - 18)/5 = -2. And y = (28 + 33)/5 = (16+9)/5 = 5. Therefore, the coordinates of B are (-2, 5). Now that we've found the coordinates of B, the next step is to find the slope of AC. Remember, the slope is calculated as the change in y divided by the change in x. The formula is (y2 - y1) / (x2 - x1). In our case, this is (8 - 3) / (4 - (-6)) = 5/10 = 1/2. The slope of AC is 1/2. The line we are looking for is perpendicular to AC, meaning it has a negative reciprocal slope. The negative reciprocal of 1/2 is -2. That's the slope of our desired line. We have a slope and a point on the line B(-2,5). Now, we will use the point-slope form which is: y - y1 = m(x - x1). Here, m is -2 and (x1, y1) is (-2, 5). Plugging in the values we get: y - 5 = -2(x + 2). Simplifying the equation, y - 5 = -2x - 4. And finally, the equation of the straight line is: y = -2x + 1. Awesome, we did it!

Conclusion: Putting it All Together

In the end, we've successfully found the equation of the straight line perpendicular to AC and passing through point B. We did this by systematically breaking down the problem into smaller, more manageable steps. We started by understanding the given information, which included the coordinates of points A and C, the ratio in which point B divides the line segment AC, and the fact that the line we're looking for is perpendicular. We then used the section formula to find the coordinates of point B, which were crucial to determining the equation of the line. After finding the coordinates of B, we calculated the slope of AC, and then used the negative reciprocal to find the slope of the perpendicular line. Finally, we used the point-slope form of the line equation to find the equation of the line. Throughout this process, we applied key concepts of coordinate geometry such as the section formula, slope calculation, and point-slope form. By following these steps, you can solve similar problems involving lines, points, and perpendicularity. Remember, the key is to break down the problem, identify the relevant formulas and concepts, and apply them systematically. Practice is key, so try solving similar problems with different coordinates and ratios. You can also explore variations of this problem, such as finding the distance from a point to a line or finding the area of the triangle formed by the points. Keep practicing, and you'll become more confident in tackling these types of geometry problems. Always remember to draw a diagram to visualize the problem and double-check your calculations. It's a great exercise to solidify your understanding and improve your problem-solving skills in math. Great job, everyone! Keep up the amazing work.