Finding Perpendicular Lines: A Step-by-Step Guide
Hey everyone, let's dive into a classic math problem: finding the equation of a line perpendicular to another and passing through a specific point. This is a fundamental concept in coordinate geometry, and understanding it unlocks a bunch of other related topics. We will break down how to solve this problem step-by-step so that anyone can understand it! Let's get started, shall we?
Understanding Perpendicular Lines
Perpendicular lines are lines that intersect at a 90-degree angle. This crucial property gives rise to a specific relationship between their slopes. The slope of a line, often represented by the letter m, describes its steepness and direction. It's calculated as the "rise over run" – the change in the y-coordinate divided by the change in the x-coordinate. When two lines are perpendicular, their slopes are negative reciprocals of each other. This means if one line has a slope of m, the slope of a perpendicular line will be -1/m. For example, if a line has a slope of 2, any line perpendicular to it will have a slope of -1/2. If a line has a slope of -3/4, a perpendicular line has a slope of 4/3. This concept is the cornerstone of solving this type of problem.
To really get this, think about the angle between the lines. If they're at right angles, the slopes have to be related in this special way. It's like a mathematical dance, where the slopes are partners, always in a specific relationship to each other. This negative reciprocal relationship ensures that the lines meet at that perfect 90-degree angle. Without this understanding, we would not be able to solve the problem and would be totally lost. Understanding perpendicular lines and their relationship with the slope is the key to solving the equation of a perpendicular line and finding the correct answer. So, the concept is the key takeaway, remember that!
Key Takeaway: The slopes of perpendicular lines are negative reciprocals of each other. This is the foundation of our calculations.
The Problem: Setting the Stage
Let's get down to the problem. We want to find the equation of a line that is perpendicular to a given line and passes through the point (5, 3). Remember, we need to know the initial line to figure out the perpendicular one. Although the exact equation of the initial line is not provided in this scenario, we can go through each of the options to find the correct answer! Each of the provided options, which are essentially equations of the line in the form of Ax + By = C, has a certain slope that can be calculated to determine if it is the correct answer. Let's analyze the options:
We need to find an equation that fulfills these two requirements. If you do not remember how to calculate the slope for each, this might be a little confusing. Keep in mind that understanding the point-slope form and the slope-intercept form of a linear equation can be super helpful here. Remember, the point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. The slope-intercept form, y = mx + b, is also useful because it directly shows the slope (m) and y-intercept (b). Keep this in mind as we analyze the options!
Key Takeaway: We're looking for a line that meets two criteria: perpendicularity and passing through the point (5, 3). We will be analyzing each of the options, let's go.
Analyzing the Options
Let's analyze the multiple choices provided. We'll examine each option, determine its slope, and check if it's perpendicular to the original line (which we need to find, using our knowledge of the slope of the perpendicular line!). We will also need to check if the line passes through the point (5, 3). Let's get to it!
Option A: 4x - 5y = 5
To find the slope, we can rewrite this equation in slope-intercept form (y = mx + b). Rearranging the terms, we get:
5y = 4x - 5
y = (4/5)x - 1
The slope of this line is 4/5. The slope of a perpendicular line would be -5/4. Next, let's see if the point (5, 3) satisfies the equation 4x - 5y = 5:
4(5) - 5(3) = 20 - 15 = 5
It does satisfy the equation! So, this option could potentially be the correct answer. Now, let's check the other options just to be sure.
Option B: 5x + 4y = 37
Converting to slope-intercept form:
4y = -5x + 37
y = (-5/4)x + 37/4
The slope is -5/4. The slope of a perpendicular line would be 4/5. Let's test if (5, 3) satisfies 5x + 4y = 37:
5(5) + 4(3) = 25 + 12 = 37
This also works! However, since the slope of the equation in B is -5/4 and not -5/4, it is not perpendicular.
Option C: 4x + 5y = 5
Converting to slope-intercept form:
5y = -4x + 5
y = (-4/5)x + 1
The slope is -4/5. The slope of a perpendicular line would be 5/4. Let's check if the point (5, 3) satisfies this equation:
4(5) + 5(3) = 20 + 15 = 35 which is not 5.
So it does not satisfy the equation. Therefore, this option is incorrect.
Option D: 5x - 4y = 8
Converting to slope-intercept form:
-4y = -5x + 8
y = (5/4)x - 2
The slope is 5/4. The slope of a perpendicular line would be -4/5. Let's check if the point (5, 3) satisfies this equation:
5(5) - 4(3) = 25 - 12 = 13 which is not 8.
So it does not satisfy the equation. Therefore, this option is incorrect.
Key Takeaway: By analyzing each option and checking for both perpendicularity and the point (5, 3), we can identify the correct answer.
Finding the Answer
So, after all that work, what's the answer? After analyzing the options and their slopes, it's pretty clear. Let's recap:
- Option A has a slope of 4/5, which is not perpendicular.
- Option B has a slope of -5/4, is perpendicular and the point works!
- Option C and D are not perpendicular.
Therefore, the answer is B. 5x + 4y = 37
Conclusion: You Got This!
Guys, finding the equation of a perpendicular line is all about understanding the relationship between slopes and applying that knowledge step-by-step. Remember the negative reciprocal rule, convert the equations to slope-intercept form, and check if the point satisfies the equation. Keep practicing, and these problems will become a breeze! You've got this, and with practice, you'll become a pro at finding the equation of a perpendicular line! Awesome work, and keep up the great work! Always remember the relationship between slopes of perpendicular lines, and you'll be set to ace these types of questions. Best of luck!