Find The Perpendicular Line Equation Through A Point

by Andrew McMorgan 53 views

Hey guys! Today, we're diving deep into the awesome world of coordinate geometry, specifically tackling a super common problem: finding the equation of a line that's perpendicular to another line and passes through a specific point. This stuff is crucial for everything from understanding graphs to more complex math problems, so let's break it down.

Understanding Perpendicular Lines

Alright, so first things first, what does it mean for two lines to be perpendicular? Basically, they intersect at a perfect 90-degree angle. In the realm of algebra and coordinate planes, this has a really neat mathematical property. If you have two lines, say line 1 with slope m1m_1 and line 2 with slope m2m_2, they are perpendicular if and only if the product of their slopes is -1. That is, m1imesm2=−1m_1 imes m_2 = -1. Another way to think about this is that the slope of one line is the negative reciprocal of the slope of the other line. So, if one line has a slope of, let's say, 2, the line perpendicular to it will have a slope of - rac{1}{2}. If a line has a slope of - rac{3}{5}, its perpendicular counterpart will have a slope of rac{5}{3}. Easy peasy, right?

Now, let's look at the given equation: $y = -4x + 9$. This equation is in the slope-intercept form, which is super handy because it directly tells us the slope and the y-intercept. In this case, the slope of this line (m1m_1) is -4, and the y-intercept is 9. We're looking for a line that is perpendicular to this one. Using our rule of negative reciprocals, the slope of our new line (m2m_2) will be the negative reciprocal of -4. So, m_2 = - rac{1}{-4} = rac{1}{4}. This means all the possible perpendicular lines will have a slope of rac{1}{4}. Keep this number rac{1}{4} in your back pocket; it's going to be super important!

Using the Point-Slope Form

So, we've figured out the slope of our target line is rac{1}{4}. But a line is defined by its slope and a point it passes through. We're given that our perpendicular line must pass through the point (4,5)(4, 5). This is where another awesome tool in coordinate geometry comes in handy: the point-slope form of a linear equation. The point-slope form is written as $y - y_1 = m(x - x_1)$, where mm is the slope of the line, and (x1,y1)(x_1, y_1) is a point on the line. It's called point-slope because, well, it uses a point and the slope to define the line!

In our problem, we know the slope m = rac{1}{4} and the point (x1,y1)=(4,5)(x_1, y_1) = (4, 5). Now, we just need to plug these values into the point-slope formula. So, we get: $y - 5 = rac{1}{4}(x - 4)$. This is a perfectly valid equation for the line we're looking for! However, usually, we want to express our final answer in the slope-intercept form ($y = mx + b$) so we can easily compare it with the given options.

To convert our point-slope equation into slope-intercept form, we just need to do a little algebraic manipulation. First, distribute the rac{1}{4} on the right side of the equation: $y - 5 = rac1}{4}x - rac{1}{4} imes 4$. Simplifying the right side gives us $y - 5 = rac{1{4}x - 1$.

Our next step is to isolate yy. We can do this by adding 5 to both sides of the equation: $y = rac1}{4}x - 1 + 5$. And finally, combine the constants on the right side $y = rac{1{4}x + 4$. Boom! We have our equation in slope-intercept form.

Checking the Options

Now that we've derived the equation of the perpendicular line, $y = rac{1}{4}x + 4$, let's compare it to the options provided:

A. $y= rac{1}{4} x+4$ B. $y= rac{1}{4} x+5$ C. $y= rac{1}{4} x+6$ D. (This option is missing in the prompt, but assuming it would be a different equation)

As you can see, our calculated equation perfectly matches option A. The slope is indeed rac{1}{4}, which is the negative reciprocal of -4, confirming it's perpendicular to the original line. And since we used the point (4,5)(4, 5) in our derivation, we know the line passes through that point. You can even double-check by plugging x=4x=4 into our equation: $y = rac{1}{4}(4) + 4 = 1 + 4 = 5$. This confirms that the point (4,5)(4, 5) lies on the line. Pretty cool, huh?

Why This Matters

Understanding how to find perpendicular lines and their equations is a fundamental skill in mathematics. It's not just about solving textbook problems; these concepts are the building blocks for more advanced topics like calculus, linear algebra, and even physics. For instance, in calculus, the concept of perpendicularity is related to the idea of orthogonal functions. In geometry, it's key to understanding shapes, distances, and areas. When you're graphing functions or analyzing data, knowing about perpendicular lines can help you spot relationships and patterns you might otherwise miss. It's all about building a strong foundation, and mastering this type of problem is a great step in that direction.

So, remember the key takeaways: perpendicular lines have slopes that are negative reciprocals of each other (m1imesm2=−1m_1 imes m_2 = -1), and the point-slope form ($y - y_1 = m(x - x_1)$) is your best friend when you have a slope and a point. Keep practicing these problems, guys, and you'll be a coordinate geometry whiz in no time! Don't be afraid to go back and re-work examples or try out variations. The more you practice, the more intuitive these concepts will become. And remember, math is like a puzzle, and each concept you learn is a new piece that helps you see the bigger picture more clearly. Keep exploring, keep questioning, and most importantly, keep enjoying the journey of learning!