Line Equation: Perpendicular To FG, Passes Through (-5, -6)

Hey math enthusiasts! Ever wondered how to find the equation of a line that's not just passing through a specific point, but also standing tall and perpendicular to another line segment? It's a classic problem in coordinate geometry, and today, we're diving deep into it. We'll break it down step by step, making sure you've got a solid grasp on the concepts. So, let's get started and unravel this mathematical puzzle together!

Understanding the Problem

Okay, let's break down this problem like a math detective cracking a case! We're on the hunt for the equation of a line, but not just any line. This line has two special clues:

1. It's like a VIP, passing right through the point (-5, -6).
2. It's standing perpendicular, making a perfect 90-degree angle, with another line segment we'll call FG. This line segment has endpoints at F(-2, -9) and G(1, -5).

So, our mission, should we choose to accept it (and we do!), is to find that equation. To do this, we'll need to channel our inner mathematicians and use some key concepts from coordinate geometry. We'll need to figure out the slope of FG, use that to find the slope of our mystery perpendicular line, and then use the point-slope form to nail down the equation. Think of it as a mathematical treasure hunt – each step gets us closer to the final answer! The line's equation will tell us everything about its behavior on the coordinate plane, its tilt, and its position. It’s the mathematical fingerprint of the line. We have to decode this fingerprint to find the equation. Let's grab our tools – our formulas, our knowledge, and our problem-solving spirit – and get to work! It might sound complicated, but don't worry, we will break it down and make it easy to understand. Are you ready to become equation hunters? Let's go! We'll take our time and solve this together. We’ll make sure to keep it fun and interesting.

Finding the Slope of FG

Alright, team, our first task is to find the slope of the line segment FG. Think of slope as the line's steepness – how much it rises or falls for every step forward. Remember that classic formula? Slope (often shown as m) is the "rise over run," which we can write as:

m = (y₂ - y₁) / (x₂ - x₁)

Where (x₁, y₁) and (x₂, y₂) are the coordinates of two points on the line. In our case, those points are F(-2, -9) and G(1, -5). So, let's plug those numbers in like pros:

m = (-5 - (-9)) / (1 - (-2))

Simplify those double negatives and do the math:

m = (-5 + 9) / (1 + 2) m = 4 / 3

Boom! The slope of FG is 4/3. That means for every 3 units we move to the right along the line, we move 4 units up. This slope is like the line segment's unique fingerprint, telling us its exact direction. But we're not done yet! Remember, we need the slope of a line perpendicular to FG. This is where things get interesting. Perpendicular lines have a special relationship when it comes to their slopes. Their slopes are what we call "negative reciprocals" of each other. This means we flip the fraction and change the sign. Knowing the slope of FG is 4/3 is a crucial stepping stone. It's like finding a vital clue in our mathematical investigation. With this in hand, we're one step closer to cracking the case of the mystery line equation. Stay with us, we're about to turn this knowledge into finding the slope of our target perpendicular line. We’re almost there!

Determining the Perpendicular Slope

Okay, now that we've nailed down the slope of FG as 4/3, it's time to unlock the secret of perpendicular slopes. Remember, perpendicular lines meet at a perfect 90-degree angle, and their slopes have a special relationship: they are negative reciprocals of each other. This might sound like a mouthful, but it's actually a pretty cool trick. So, what does "negative reciprocal" mean? It's like a double transformation for our slope. First, we flip the fraction. So, 4/3 becomes 3/4. Then, we change the sign. Since 4/3 was positive, 3/4 becomes negative. Voila! The slope of our line perpendicular to FG is -3/4. This negative reciprocal relationship is a mathematical shortcut, a neat little trick that saves us from having to calculate angles and tangents. It's a direct link between the slopes of perpendicular lines. Think of it like this: if one line is climbing steeply uphill (positive slope), a line perpendicular to it must be going downhill (negative slope) and at a complementary angle (reciprocal). This new slope, -3/4, is a crucial piece of the puzzle. It tells us the steepness and direction of our target line, the one that passes through (-5, -6) and forms that perfect 90-degree angle with FG. It’s a unique attribute of our line, like its DNA. And now that we know it, we're ready to take the next step towards finding the full equation of the line. We're on a roll! Let’s keep the momentum going and use this perpendicular slope to finally write the equation of our mystery line.

Using Point-Slope Form

Fantastic! We've got the slope of our perpendicular line: -3/4. Now, let's put this knowledge to work and find the actual equation of the line. This is where the point-slope form comes in handy. It's a super useful way to write the equation of a line when you know its slope and one point it passes through. The point-slope form looks like this:

y - y₁ = m(x - x₁)

Where:

• m is the slope (which we know is -3/4).
• (x₁, y₁) is the point the line passes through (which we know is (-5, -6)).

Time to plug in those values and watch the magic happen! Substitute -3/4 for m, -5 for x₁, and -6 for y₁:

y - (-6) = -3/4(x - (-5))

Let's simplify those double negatives:

y + 6 = -3/4(x + 5)

This is the equation of our line in point-slope form! It's technically correct, but often, we want to see the equation in slope-intercept form (y = mx + b) or standard form (Ax + By = C). So, let's keep going and transform this equation into a form that's even easier to read and use. This point-slope form is like a partially decoded message. It contains all the necessary information, but it's not in its most polished, easy-to-understand form yet. We are like codebreakers, transforming it into something everyone can read. We've done the hard part of finding the slope and using the point-slope formula. Now it's just a bit of algebraic maneuvering to get it into a more common format. Are you ready to see the final form of our equation? Let's move on and finish the job!

Converting to Slope-Intercept Form

Alright, we've got our equation in point-slope form: y + 6 = -3/4(x + 5). That's a great start, but let's make it even clearer by converting it to slope-intercept form. This form, y = mx + b, is super popular because it tells us the slope (m) and the y-intercept (b) at a glance. It's like the line's calling card, giving us its key characteristics instantly. To get there, we need to do a little algebraic maneuvering. Our goal is to isolate y on the left side of the equation. First, let's distribute the -3/4 on the right side:

y + 6 = -3/4x - 15/4

Now, we need to get rid of that +6 on the left. We can do that by subtracting 6 from both sides. But remember, to subtract 6 from -15/4, we need a common denominator. So, let's rewrite 6 as 24/4:

y + 6 - 6 = -3/4x - 15/4 - 24/4

Simplify it down:

y = -3/4x - 39/4

There it is! Our equation is now in slope-intercept form. We can see that the slope (m) is -3/4 (which we already knew!) and the y-intercept (b) is -39/4. This means the line crosses the y-axis at the point (0, -39/4). This slope-intercept form is like a clear, concise summary of the line's behavior. It tells us the line's steepness, direction, and where it intersects the y-axis. It’s a powerful form that allows us to quickly visualize and understand the line. We've transformed our equation from a good start in point-slope form to this easily readable and informative slope-intercept form. We are really close to presenting the final equation! Let's recap what we've done and see if we can take it one step further, maybe even to standard form.

The Final Equation

Alright, guys, after our mathematical journey, we've arrived at the equation of the line that passes through (-5, -6) and is perpendicular to FG:

y = -3/4x - 39/4

This is the slope-intercept form, and it's a fantastic way to represent the line. It clearly shows us the slope (-3/4) and the y-intercept (-39/4). But, just for kicks, let's take it one step further and convert it to standard form. Standard form is written as Ax + By = C, where A, B, and C are integers, and A is usually positive. To get there, let's first get rid of the fraction by multiplying every term in our equation by 4:

4y = -3x - 39

Now, let's move the -3x term to the left side by adding 3x to both sides:

3x + 4y = -39

And there we have it! The equation of our line in standard form is:

3x + 4y = -39

We did it! We started with a geometric challenge, navigated through slopes and points, and ended up with a beautiful equation that represents our line. This equation, in both slope-intercept and standard forms, is the ultimate answer. It's like the final piece of the puzzle, completing the picture of our mathematical quest. We've not only found the answer, but we've also reinforced our understanding of slopes, perpendicular lines, and different forms of linear equations. This was a real workout for our math brains, and we came out victorious! Give yourselves a pat on the back, mathletes. You've successfully tackled this problem and added another tool to your mathematical toolkit.

Conclusion

So, there you have it, friends! We've successfully navigated the world of coordinate geometry to find the equation of a line that not only passes through a specific point but also stands perpendicular to another line segment. We started by understanding the problem, then we calculated the slope of FG, used the negative reciprocal trick to find the perpendicular slope, plugged everything into the point-slope form, and finally, transformed our equation into both slope-intercept and standard forms. Whew! It was quite the journey, but we tackled it like pros.

This problem is a perfect example of how different concepts in math connect and build upon each other. We used the slope formula, the concept of perpendicular lines, the point-slope form, and algebraic manipulation, all to solve one problem. It's like a mathematical symphony, where each instrument (or concept) plays its part to create the final harmonious result. But more than just getting the right answer, we've also sharpened our problem-solving skills. We've learned to break down complex problems into smaller, manageable steps, a skill that's valuable not just in math, but in all areas of life. So, the next time you encounter a math challenge, remember this journey. Remember how we took it step by step, used our knowledge and tools, and ultimately conquered the problem. You've got this! Keep exploring, keep learning, and keep those math muscles strong. Until next time, happy equation hunting!