X-Intercept Of Perpendicular Line CD: A Math Solution

by Andrew McMorgan 54 views

Hey Plastik Magazine readers! Let's dive into a cool math problem today. We're going to figure out how to find the x-intercept of a line that's perpendicular to another line. Sounds a bit complicated? Don't worry, we'll break it down step by step. We've got line CD that’s perpendicular to line AB, and it passes through point C(5,12). We also know the coordinates of points A and B are (-10,-3) and (7,14), respectively. Our mission? Find where line CD crosses the x-axis. Let's get started!

1. Understanding Perpendicular Lines and Slopes

Okay, first things first, let's talk about perpendicular lines. Remember that two lines are perpendicular if they intersect at a right angle (90 degrees). Now, here's the key: the slopes of perpendicular lines have a special relationship. If one line has a slope of m, the slope of a line perpendicular to it is -1/m. This is super important for solving our problem. So, understanding slopes of perpendicular lines is crucial. We're dealing with lines that intersect at a perfect 90-degree angle, and that means their slopes are negative reciprocals of each other. Keep this in mind as we move forward; it's the foundation for finding our solution. Let's make sure we've got this concept down pat before we proceed. Grasping this relationship between slopes is like having the secret code to unlock this mathematical puzzle. With this understanding, we can confidently tackle the next steps and get closer to finding that x-intercept.

2. Calculating the Slope of Line AB

Now that we've refreshed our memory on perpendicular lines, let's calculate the slope of line AB. Remember the slope formula? It's (y2 - y1) / (x2 - x1). We have the coordinates of A (-10, -3) and B (7, 14). Let's plug those values into the formula: Slope of AB = (14 - (-3)) / (7 - (-10)) = (14 + 3) / (7 + 10) = 17 / 17 = 1. So, the slope of line AB is 1. This is a crucial piece of information because it will help us find the slope of line CD, which is perpendicular to AB. Calculating this slope is like setting the stage for the rest of our solution. It gives us a solid foundation to build upon. Next, we'll use this slope to find the slope of line CD, and then we'll be well on our way to finding the x-intercept. So, we've successfully calculated the slope of AB, and we're ready to move on to the next step. Keep up the great work, guys! We're making progress and getting closer to cracking this mathematical code.

3. Determining the Slope of Line CD

Alright, we've got the slope of line AB, which is 1. Now, let's find the slope of line CD. Remember, line CD is perpendicular to line AB, so its slope will be the negative reciprocal of AB's slope. Since the slope of AB is 1, the slope of CD is -1/1, which simplifies to -1. Easy peasy, right? Knowing the slope of CD is super important because it's a key ingredient in finding the equation of line CD. Think of it like this: we're building the equation of line CD piece by piece, and the slope is one of the main components. Once we have the slope and a point on the line (which we already have – point C), we can use the point-slope form to write the equation of the line. So, we've successfully found the slope of CD, and we're one step closer to our goal. Pat yourselves on the back, guys! We're doing great. Let's keep this momentum going as we move on to the next step.

4. Finding the Equation of Line CD

Now that we know the slope of line CD is -1 and it passes through point C(5, 12), we can find the equation of line CD. We'll use the point-slope form of a linear equation: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Plugging in our values, we get: y - 12 = -1(x - 5). Let's simplify this equation: y - 12 = -x + 5. Now, let's get it into slope-intercept form (y = mx + b) by adding 12 to both sides: y = -x + 17. Voila! We have the equation of line CD. Finding the equation is a big step because it gives us a complete picture of the line. We know its slope and its y-intercept (which is 17 in this case). With the equation in hand, we can now find any point on the line, including the x-intercept, which is what we're ultimately after. So, we've successfully found the equation of line CD, and we're in the home stretch now. We're just one step away from finding that x-intercept. Let's keep our focus and finish strong!

5. Calculating the X-Intercept of Line CD

Okay, guys, we're in the final stretch! We have the equation of line CD: y = -x + 17. To find the x-intercept, we need to find the point where the line crosses the x-axis. Remember, on the x-axis, y is always 0. So, we'll set y = 0 in our equation and solve for x: 0 = -x + 17. Add x to both sides: x = 17. And there you have it! The x-intercept of line CD is 17. Calculating the x-intercept was the final piece of the puzzle. We've gone from understanding perpendicular lines and slopes to finding the equation of a line and finally pinpointing where it crosses the x-axis. This is a fantastic accomplishment! We've successfully navigated this math problem, and we've learned a lot along the way. Give yourselves a big round of applause, guys! You've earned it.

Conclusion

So, to recap, we found the x-intercept of line CD to be 17. We did this by first understanding the relationship between perpendicular lines and their slopes, then calculating the slope of line AB, determining the slope of line CD, finding the equation of line CD, and finally, calculating the x-intercept. What a journey! I hope you guys found this step-by-step guide helpful. Remember, math can be fun when we break it down into manageable steps. Keep practicing, keep exploring, and keep that mathematical curiosity alive! Until next time, keep shining, Plastik Magazine readers!