Find The Perpendicular Line Equation Through A Point

Understanding how to find the equation of a line that's perpendicular to another line and passes through a specific point is a fundamental skill in coordinate geometry. It’s a concept that pops up frequently in various mathematical contexts, from graphing and calculus to more advanced fields like linear algebra. This article will guide you through the process, breaking down each step so you can confidently tackle these problems. We'll also touch on how graphing can help you visualize and verify your solution. So, let's dive in and demystify the process of finding perpendicular lines!

Understanding Perpendicular Lines and Their Slopes

Before we start finding equations, let's clarify what makes two lines perpendicular. In the realm of coordinate geometry, two non-vertical lines are perpendicular if and only if the product of their slopes is -1. This means that if you have a line with a slope $m_1$, the slope of any line perpendicular to it, let's call it $m_2$, will be the negative reciprocal of $m_1$. Mathematically, this relationship is expressed as $m_1 imes m_2 = -1$, or $m_2 = -1/m_1$. This crucial relationship is the cornerstone of solving problems involving perpendicular lines. If one line is horizontal (slope = 0), then a line perpendicular to it will be vertical (undefined slope), and vice versa. For our example, we are given the line $y = 2x + 1$. This equation is in the slope-intercept form, $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept. From this form, we can easily identify the slope of the given line. The slope of the line $y = 2x + 1$ is $m_1 = 2$. Now, we need to find the slope of the line that is perpendicular to it. Using our rule, the perpendicular slope $m_2$ will be the negative reciprocal of $m_1$. So, $m_2 = -1/2$. This means our new line, the one we need to find the equation for, will have a slope of -1/2. Remember, this is the key to unlocking the rest of the problem. Without this slope, we wouldn't know how our target line is oriented on the coordinate plane.

Using the Point-Slope Form to Find the Equation

Now that we have the slope of our perpendicular line ($m_2 = -1/2$) and a point it passes through ($P(2, 8)$), we can use the point-slope form of a linear equation. The point-slope form is written as $y - y_1 = m(x - x_1)$, where $m$ is the slope of the line, and $(x_1, y_1)$ are the coordinates of a point on the line. In our case, $m = -1/2$, $x_1 = 2$, and $y_1 = 8$. Plugging these values into the point-slope form, we get: $y - 8 = -1/2(x - 2)$. This equation accurately represents the line we are looking for. It tells us the relationship between any point $(x, y)$ on this perpendicular line and the specific point $(2, 8)$ that it must pass through, using the slope we calculated. It's a direct application of the geometric properties we discussed earlier. The point-slope form is incredibly useful because it allows us to construct the equation of a line when we know its steepness (slope) and at least one location it occupies. It’s like having a starting point and a direction – you can chart the entire course of the line from there. Many students find this form more intuitive than the slope-intercept form when they are first learning about linear equations because it explicitly uses the given point. Once we have this equation, we can also convert it into other forms, such as the slope-intercept form ($y = mx + b$) or the standard form ($Ax + By = C$), depending on what the problem requires or what is most convenient for further analysis or graphing. Let's go ahead and convert this into slope-intercept form, as it's often preferred for graphing.

Converting to Slope-Intercept Form

To convert the point-slope equation $y - 8 = -1/2(x - 2)$ into the slope-intercept form ($y = mx + b$), we need to isolate $y$. First, distribute the $-1/2$ on the right side of the equation: $y - 8 = -1/2x + (-1/2)(-2)$. This simplifies to $y - 8 = -1/2x + 1$. Now, to get $y$ by itself, we add 8 to both sides of the equation: $y = -1/2x + 1 + 8$. Finally, combine the constant terms: $y = -1/2x + 9$. So, the equation of the line passing through the point $(2, 8)$ and perpendicular to the line $y = 2x + 1$ is $y = -1/2x + 9$. This form is very helpful because it immediately tells us the slope (-1/2) and the y-intercept (9), which are essential for graphing the line. The slope confirms that it's indeed perpendicular to the original line, and the y-intercept gives us another point on the line, which is useful for plotting. This conversion process is straightforward algebraic manipulation, and mastering it allows you to present your linear equations in the most useful format for different applications.

Graphing to Verify Perpendicularity

Visual confirmation is a powerful tool in mathematics, and graphing our lines is an excellent way to check if they are indeed perpendicular. We have our original line, $y = 2x + 1$, and our newly found perpendicular line, $y = -1/2x + 9$. Let's plot these on a coordinate plane. For the original line, $y = 2x + 1$: The y-intercept is at $(0, 1)$. The slope is 2, meaning for every 1 unit we move to the right on the x-axis, we move 2 units up on the y-axis. So, from $(0, 1)$, we can find another point by moving 1 unit right and 2 units up, landing us at $(1, 3)$. Plotting these two points $(0, 1)$ and $(1, 3)$ and drawing a line through them gives us the graph of $y = 2x + 1$. Now, for our perpendicular line, $y = -1/2x + 9$: The y-intercept is at $(0, 9)$. The slope is -1/2, meaning for every 2 units we move to the right on the x-axis, we move 1 unit down on the y-axis. From the y-intercept $(0, 9)$, we can find another point by moving 2 units right and 1 unit down, landing us at $(2, 8)$. Notice that this point $(2, 8)$ is the specific point $P$ we were given, which is a good sign! Plotting the points $(0, 9)$ and $(2, 8)$ and drawing a line through them gives us the graph of $y = -1/2x + 9$. When you look at the two graphs together, you should clearly see that they intersect at a right angle. The steepness and direction of the lines will visually demonstrate their perpendicular relationship. One line goes up and to the right relatively sharply (slope of 2), while the other goes down and to the right more gradually (slope of -1/2). The visual confirmation is often the most satisfying part of solving these problems, as it makes the abstract concept of perpendicularity tangible. If your lines don't appear to intersect at a 90-degree angle, it’s a strong indication that there might have been a calculation error, prompting you to revisit your steps. This visual check reinforces the algebraic solution and builds confidence in your understanding.

Common Pitfalls and How to Avoid Them

When working with perpendicular lines, there are a few common mistakes that can trip you up. One of the most frequent errors is in calculating the slope of the perpendicular line. Remember, it’s not just the negative of the original slope; it's the negative reciprocal. So, if the original slope is $m$, the perpendicular slope is $-1/m$. A student might mistakenly think the perpendicular slope to $y=2x+1$ (slope 2) is $-2$, but it should be $-1/2$. Always double-check that you've taken both the negative sign and the reciprocal. Another common issue arises when the original line is given in a form other than slope-intercept. If you're given an equation like $3x + 4y = 12$, you first need to convert it to $y = mx + b$ form to easily identify the slope. Solving for $y$ would give $4y = -3x + 12$, and then $y = -3/4x + 3$. The slope is $-3/4$. Then, the perpendicular slope would be $4/3$. Always ensure you've correctly isolated $y$ to find the slope accurately. Misplacing the given point $(x_1, y_1)$ in the point-slope formula is also a common pitfall. Make sure $x_1$ is substituted with the x-coordinate and $y_1$ with the y-coordinate. Forgetting to distribute the slope in the point-slope form, or making errors during algebraic simplification, can lead to an incorrect final equation. Carefully distributing the slope and meticulously performing each algebraic step will help prevent these errors. Finally, while graphing is a great verification tool, it's not a substitute for accurate calculations. A poorly drawn graph can sometimes make it difficult to definitively confirm perpendicularity, especially if the slopes are very close in magnitude. Always rely on your calculated slopes and the negative reciprocal rule first, and then use the graph as a visual aid to reinforce your findings. By being mindful of these common errors and following the steps carefully, you can ensure accurate and reliable results when finding perpendicular line equations.

Conclusion

Finding the equation of a line perpendicular to a given line and passing through a specific point involves a clear, step-by-step process. We first identified the slope of the given line ($m_1 = 2$). Then, we calculated the slope of the perpendicular line by taking the negative reciprocal ($m_2 = -1/2$). Using the point-slope form, $y - y_1 = m(x - x_1)$, with the given point $P(2, 8)$ and the perpendicular slope, we formulated the equation $y - 8 = -1/2(x - 2)$. Finally, we converted this into the slope-intercept form, $y = -1/2x + 9$, which is easy to graph. Visualizing these lines on a coordinate plane confirms their perpendicular relationship. Mastering this concept not only solidifies your understanding of linear equations but also equips you with a valuable tool for tackling more complex geometric problems. Remember the key: the product of the slopes of perpendicular lines is -1. For further exploration into coordinate geometry and linear equations, you can refer to resources like Khan Academy's excellent tutorials on the subject.