Finding The Equation Of A Line At 45 Degrees A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem that involves finding the equation of a line. This isn't just any line; it's a line that makes a 45-degree angle with another given line and passes through a specific point. Sounds intriguing, right? Let's break it down step by step and conquer this geometric challenge!

Understanding the Problem

Our main goal here is to determine the equation of a line. We know this line passes through the point (1, 2) and forms a 45-degree angle with the line 2x + 3y - 10 = 0. The problem might seem complex at first glance, but don't worry! We'll use our knowledge of slopes, angles, and line equations to solve it. The key here is understanding how the angle between two lines relates to their slopes. We'll also need to remember the point-slope form of a line equation, which will be super handy given that we have a point the line passes through.

Before we jump into calculations, let's visualize what we're trying to find. Imagine a coordinate plane with the line 2x + 3y - 10 = 0 drawn on it. Now, picture a point at (1, 2). There are actually two lines that can pass through (1, 2) and form a 45-degree angle with the original line. Our mission is to find the equations for both of these lines. This involves a bit of algebraic manipulation and a solid understanding of trigonometric relationships, specifically the tangent function, which connects angles and slopes. So, buckle up, and let's get started on this exciting journey to find these elusive line equations!

Key Concepts and Formulas

Before we dive into the solution, let's quickly recap some key concepts and formulas that we'll be using. These are the building blocks that will help us solve this problem efficiently and accurately.

First, let's talk about the slope of a line. The slope, often denoted by 'm', tells us how steep a line is. Mathematically, it's the change in the y-coordinate divided by the change in the x-coordinate. Given a line in the form Ax + By + C = 0, the slope can be calculated as m = -A/B. This is a crucial formula for us because we're given the equation 2x + 3y - 10 = 0, and we'll need its slope to proceed.

Next up, we need to understand the angle between two lines. If we have two lines with slopes m1 and m2, the tangent of the angle (θ) between them is given by the formula: tan(θ) = |(m1 - m2) / (1 + m1 * m2)|. Notice the absolute value signs! This is because the angle between two lines can be acute or obtuse, and we're primarily concerned with the magnitude of the angle in this case. Since we know the angle is 45 degrees, we know tan(45°) = 1, which will simplify our calculations.

Finally, we'll use the point-slope form of a line equation. This form is super useful when we know a point (x1, y1) on the line and its slope (m). The equation is: y - y1 = m(x - x1). We know our line passes through the point (1, 2), so we already have (x1, y1). Once we find the possible slopes of our line, we can easily plug these values into the point-slope form to get the equations of the lines.

By mastering these concepts – slope calculation, the angle between two lines formula, and the point-slope form – we're well-equipped to tackle this problem. These are fundamental tools in coordinate geometry, and understanding them deeply will not only help with this problem but also with many other geometric challenges.

Solving the Problem Step-by-Step

Okay, let's get down to business and solve this problem step-by-step. We're going to break it down into manageable chunks to make sure we don't miss anything. Remember, our goal is to find the equations of the lines that pass through (1, 2) and make a 45-degree angle with the line 2x + 3y - 10 = 0.

Step 1: Find the slope of the given line. We have the line 2x + 3y - 10 = 0. Using the formula m = -A/B, where A is the coefficient of x and B is the coefficient of y, we get m1 = -2/3. This is the slope of the line we're given, and it's our starting point for finding the slopes of the other lines.

Step 2: Use the angle formula to find the possible slopes. We know the angle between the lines is 45 degrees, and we know that tan(45°) = 1. Let m2 be the slope of the line we're trying to find. Using the formula for the tangent of the angle between two lines, we have: 1 = |(m2 - (-2/3)) / (1 + m2 * (-2/3))|. This looks a bit intimidating, but we'll simplify it. First, let's get rid of the absolute value by considering both positive and negative cases:

• Case 1: (m2 + 2/3) / (1 - (2/3)m2) = 1
• Case 2: (m2 + 2/3) / (1 - (2/3)m2) = -1

Now, we'll solve each case separately. This will give us two possible values for m2, which correspond to the two lines that satisfy the conditions of the problem.

Step 3: Solve for m2 in each case. Let's tackle Case 1 first. Multiplying both sides by (1 - (2/3)m2) gives us m2 + 2/3 = 1 - (2/3)m2. Now, let's get all the m2 terms on one side and the constants on the other: m2 + (2/3)m2 = 1 - 2/3. This simplifies to (5/3)m2 = 1/3. Multiplying both sides by 3/5, we get m2 = 1/5. So, one possible slope for our line is 1/5.

Now, let's move on to Case 2. Multiplying both sides by (1 - (2/3)m2) gives us m2 + 2/3 = -1 + (2/3)m2. Again, let's rearrange the terms: m2 - (2/3)m2 = -1 - 2/3. This simplifies to (1/3)m2 = -5/3. Multiplying both sides by 3, we get m2 = -5. So, our second possible slope is -5.

Step 4: Use the point-slope form to find the equations of the lines. We now have two possible slopes: m2 = 1/5 and m2 = -5. We also know that both lines pass through the point (1, 2). We'll use the point-slope form, y - y1 = m(x - x1), to find the equations of the lines.

• For m2 = 1/5: y - 2 = (1/5)(x - 1). Multiplying by 5 to eliminate the fraction, we get 5y - 10 = x - 1. Rearranging, we get the equation x - 5y + 9 = 0.
• For m2 = -5: y - 2 = -5(x - 1). Expanding, we get y - 2 = -5x + 5. Rearranging, we get the equation 5x + y - 7 = 0.

Step 5: State the final answer. We've done it! We've found the equations of the two lines that pass through (1, 2) and make a 45-degree angle with the line 2x + 3y - 10 = 0. The equations are: x - 5y + 9 = 0 and 5x + y - 7 = 0. Give yourselves a pat on the back for cracking this problem!

Common Mistakes and How to Avoid Them

Solving geometry problems like this can be tricky, and it's easy to make a few common mistakes along the way. But don't worry, guys! We're going to highlight these pitfalls and learn how to avoid them. Being aware of these potential errors can save you time and frustration, and ensure you get the correct answer.

Mistake 1: Forgetting the absolute value in the angle formula. The formula for the tangent of the angle between two lines involves an absolute value: tan(θ) = |(m1 - m2) / (1 + m1 * m2)|. This is crucial because there are two possible lines that can form a given angle with another line. Forgetting the absolute value can lead to missing one of the solutions. To avoid this, always remember to consider both the positive and negative cases when solving for the slope (m2).

Mistake 2: Incorrectly calculating the slope of the given line. The slope of a line in the form Ax + By + C = 0 is given by m = -A/B. It's easy to mix up the signs or divide the coefficients incorrectly. Double-check your calculation to ensure you have the correct slope for the given line. A simple way to verify is to rewrite the equation in slope-intercept form (y = mx + b) and compare the 'm' value.

Mistake 3: Algebraic errors while solving for the slopes. Solving the equation after applying the tangent formula involves algebraic manipulation, and it's easy to make mistakes with signs, fractions, or distribution. Take your time, write each step clearly, and double-check your work. If possible, try simplifying the equation in stages to reduce the chances of errors. For instance, clear fractions by multiplying through by a common denominator before expanding brackets.

Mistake 4: Using the wrong form of the line equation. Once you've found the slope, it's essential to use the correct form of the line equation to find the final answer. Since we have a point and the slope, the point-slope form (y - y1 = m(x - x1)) is the most convenient. Make sure you plug in the values correctly and then rearrange the equation into the desired form (e.g., Ax + By + C = 0).

Mistake 5: Not checking the answer. After you've found the equations of the lines, it's a good idea to check your answer. You can do this by plugging the point (1, 2) into your equations to make sure it satisfies them. You can also visualize the lines and the given line on a graph to see if the angles look approximately correct. This simple check can help you catch any errors and give you confidence in your solution.

By being mindful of these common mistakes and taking steps to avoid them, you'll be well on your way to mastering these types of geometry problems. Remember, practice makes perfect, so keep working at it!

Real-World Applications of Line Equations and Angles

Now that we've tackled this interesting problem, let's take a step back and think about why this kind of math is actually useful. You might be wondering, "Where in the real world would I ever need to find the equation of a line that makes a certain angle with another line?" Well, guys, the applications are more common than you might think!

Architecture and Engineering: In architecture and engineering, precise angles and line equations are crucial for designing structures. For example, when designing a roof, engineers need to calculate the angle of the roofline to ensure proper drainage and structural integrity. This involves working with line equations and angles, just like in our problem. Similarly, when planning the layout of a building, architects use coordinate geometry to determine the placement of walls, windows, and doors, often needing to calculate angles and intersections.

Navigation: Navigation systems, whether in ships, airplanes, or even your car's GPS, rely heavily on coordinate geometry and the relationships between lines and angles. Calculating the course of a ship or airplane involves determining the angle relative to a certain direction (e.g., north) and using line equations to represent the path of travel. Understanding how to work with angles and line equations is essential for accurate navigation.

Computer Graphics and Game Development: In computer graphics and game development, lines and angles are fundamental for creating realistic images and animations. When creating 3D models or animating objects, developers use mathematical equations to define the shapes, positions, and movements of objects. Calculating angles and line equations is essential for creating accurate perspectives, lighting effects, and object interactions.

Robotics: Robotics involves programming robots to perform specific tasks, which often requires precise movements and positioning. Robots use coordinate systems and mathematical equations to navigate their environment, manipulate objects, and avoid obstacles. Calculating angles and line equations is crucial for programming robots to perform tasks such as welding, painting, or assembling products.

Physics: Physics is full of applications for line equations and angles. For example, when studying projectile motion, physicists use equations to describe the trajectory of an object, which is essentially a line or curve in space. Calculating the angle of launch and the initial velocity is essential for predicting the range and height of a projectile. Similarly, when studying optics, understanding the angles at which light rays reflect and refract is crucial for designing lenses and optical instruments.

As you can see, the concepts we've explored in this problem have wide-ranging applications in various fields. From designing buildings to navigating ships to creating video games, the ability to work with lines, angles, and their equations is a valuable skill. So, keep practicing and honing your math skills – you never know where they might take you!

Conclusion

Wow, guys! We've really taken a deep dive into the world of lines, angles, and equations. We started with a challenging problem – finding the equations of lines that pass through a specific point and form a 45-degree angle with another line. We broke down the problem step-by-step, revisiting key concepts like slope, the angle between lines formula, and the point-slope form of a line equation. We tackled the algebraic challenges, solved for the slopes, and found the equations of the two lines that satisfied the conditions. It was quite the mathematical workout!

But we didn't stop there. We also explored common mistakes that people make when solving these types of problems and learned how to avoid them. Being aware of these pitfalls – like forgetting the absolute value in the angle formula or making algebraic errors – can significantly improve your problem-solving accuracy and efficiency. Remember, math is like any other skill; the more you practice and the more mindful you are of potential errors, the better you'll become.

Finally, we broadened our perspective and looked at the real-world applications of line equations and angles. We saw how these concepts are fundamental in fields like architecture, engineering, navigation, computer graphics, robotics, and physics. From designing buildings to programming robots, the ability to work with lines and angles is a valuable asset.

So, what's the takeaway from all of this? Math isn't just about memorizing formulas and solving equations. It's about developing critical thinking skills, problem-solving abilities, and an understanding of the world around us. The next time you encounter a geometry problem, remember the steps we've discussed, the common mistakes to avoid, and the real-world relevance of the concepts. And most importantly, don't be afraid to challenge yourself and explore the fascinating world of mathematics!