Line Perpendicular To BC Passing Through A: Step-by-Step Guide

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Hey guys! Today, we are diving into a cool geometry problem. We've got a triangle ABC, defined by the points A(3,8), B(7,5), and C(2,3). Our mission, should we choose to accept it (and we totally do!), is to find the equation of a line that passes right through point A and is perpendicular to the side BC. Sounds like a fun challenge, right? Let's break it down and make it super clear how to tackle this kind of problem. So buckle up, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into calculations, let’s make sure we understand exactly what we're trying to find. We need a line. Remember, the equation of a line usually looks something like y = mx + b, where m is the slope and b is the y-intercept. We already know the line has to pass through point A(3,8). That's a great start! But there are infinitely many lines that pass through a single point. What makes this line special is that it must be perpendicular to side BC of our triangle. Ah, that gives us a crucial piece of information about its direction! So, we need to figure out how to use the fact that our line is perpendicular to BC to find its slope (m) and then use point A to find the y-intercept (b). This is a classic geometry problem that combines coordinate geometry and the properties of perpendicular lines. We've got this!

Step 1: Finding the Slope of BC

Okay, first things first, let’s figure out the slope of the line segment BC. Remember the slope formula? It’s rise over run, or the change in y divided by the change in x. Mathematically, that looks like this:

m = (y₂ - y₁) / (x₂ - x₁)

We have the coordinates for B(7,5) and C(2,3). Let's plug those into our formula. We can call B our (x₁, y₁) and C our (x₂, y₂). So we get:

m_BC = (3 - 5) / (2 - 7) = -2 / -5 = 2/5

Alright! The slope of BC (m_BC) is 2/5. That means for every 5 units we move to the right along BC, we move 2 units up. This slope is crucial because it's directly related to the slope of the line we're actually trying to find. We are making progress, guys!

Step 2: Finding the Slope of the Perpendicular Line

Now for the cool part: the relationship between perpendicular lines. Here’s the key: If two lines are perpendicular, their slopes are negative reciprocals of each other. What does that mean in plain English? It means you flip the fraction and change the sign. So, if the slope of BC is 2/5, the slope of a line perpendicular to BC is -5/2. Let's call the slope of our perpendicular line m_perp. So:

m_perp = -1 / m_BC = -1 / (2/5) = -5/2

There we go! The slope of the line we're trying to find is -5/2. That’s a pretty steep negative slope, which makes sense if it's perpendicular to BC, which has a positive slope. Understanding this negative reciprocal relationship is fundamental in coordinate geometry. Feels good to nail this, right?

Step 3: Using Point-Slope Form

We've got the slope of our line (m_perp = -5/2), and we know it passes through point A(3,8). Now we can use the point-slope form of a line equation. This is a super handy form to know! It looks like this:

y - y₁ = m(x - x₁)

Where (x₁, y₁) is a point on the line, and m is the slope. We know m and we know (x₁, y₁) – that's point A! Let's plug in our values:

y - 8 = (-5/2)(x - 3)

Awesome! We’ve got an equation for our line. The point-slope form is a powerful tool because it allows us to quickly write the equation of a line if we know a point and the slope. But, we aren’t quite done yet. Let's clean this up and put it in slope-intercept form.

Step 4: Converting to Slope-Intercept Form

Most of the time, we like our line equations in slope-intercept form, y = mx + b. So, let’s take the equation we got from the point-slope form and rearrange it. First, we distribute the -5/2:

y - 8 = (-5/2)x + (15/2)

Now, we add 8 to both sides. But, we need to add 8 in the form of a fraction with a denominator of 2, so we write 8 as 16/2:

y = (-5/2)x + (15/2) + (16/2)

Combine those fractions, and we get:

y = (-5/2)x + 31/2

Boom! There it is! Our equation is in slope-intercept form. Converting between different forms of linear equations is a key skill in algebra and geometry. We are mastering this step-by-step!

Step 5: The Final Equation

So, the equation of the line that passes through point A(3,8) and is perpendicular to BC is:

y = (-5/2)x + 31/2

This tells us that the line has a slope of -5/2 and a y-intercept of 31/2 (which is 15.5). You could even graph this line and triangle ABC to visually check that it all makes sense. Go ahead and try it! Visualizing the problem often helps solidify your understanding. This is so satisfying when everything clicks into place!

Alternative Form: Standard Form

Just for kicks, let's also convert our equation to standard form. Standard form looks like Ax + By = C, where A, B, and C are integers, and A is usually positive. To get our equation into standard form, we want to eliminate the fraction. Multiply the entire equation by 2:

2y = -5x + 31

Now, add 5x to both sides:

5x + 2y = 31

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

5x + 2y = 31

Knowing multiple forms of a linear equation is super useful because sometimes one form is more convenient than another depending on the problem. We are becoming equation ninjas!

Key Takeaways

Wow, we did it! We successfully found the equation of a line that passes through a given point and is perpendicular to another line segment. Let's recap the key steps we took:

  1. Found the slope of BC using the slope formula.
  2. Found the slope of the perpendicular line by taking the negative reciprocal of the slope of BC.
  3. Used point-slope form to write the equation of the line.
  4. Converted to slope-intercept form to get the equation in the familiar y = mx + b format.
  5. (Bonus!) Converted to standard form to see yet another way to represent the line.

Mastering these steps opens the door to solving a wide range of geometry problems. You now have another tool in your mathematical toolbox!

Practice Makes Perfect

The best way to really understand this stuff is to practice! Try changing the coordinates of points A, B, and C and going through the same steps. Or, try finding the equation of a line parallel to BC that passes through A. Remember, parallel lines have the same slope. So many possibilities! Consistent practice builds confidence and deepens your understanding. Keep at it, guys!

Conclusion

So, that’s how you find the equation of a line perpendicular to a side of a triangle and passing through a given point. This problem touches on so many fundamental concepts in coordinate geometry – slope, perpendicular lines, point-slope form, slope-intercept form, standard form. By working through this problem, you've strengthened your understanding of all these concepts. I hope this breakdown was clear and helpful. Remember, math is a journey, not a destination. Keep exploring, keep learning, and most importantly, keep having fun with it! You've got this!

If you have any questions or want to try another example, feel free to ask! Until next time, happy calculating!