Calculating Distance Between Parallel Lines R And S

Hey guys! Today, we're diving into a fun problem from the realm of mathematics: calculating the distance between two parallel lines. Specifically, we're given the equations of two lines, line r and line s, and our mission is to find the distance that separates them. The equations are: r: x + 3y - 10 = 0 and s: x + 3y - 6 = 0. Sounds like a geometric adventure, right? Let's break it down step by step and make sure we not only get to the answer but also understand the 'why' behind it. So grab your thinking caps, and let’s get started!

Understanding the Problem

Before we jump into calculations, let's take a moment to visualize what we're dealing with. We have two lines, r and s, defined by their equations. Notice anything similar about these equations? That's right, the coefficients of x and y are the same (1 and 3, respectively). This is a crucial observation because it tells us that the lines are parallel. Parallel lines, as you might remember, never intersect, and they maintain a constant distance from each other. This constant distance is what we're trying to find.

Now, why is this important? Well, imagine trying to measure the distance between lines that aren't parallel. The distance would change depending on where you measure it! But because our lines are parallel, we can pick any point on one line and find the shortest distance to the other line, and that will be the distance between the lines. Think of it like measuring the width of a lane on a highway – it’s consistent all the way down.

The challenge now is to figure out how to actually calculate this distance. We could try graphing the lines and measuring, but that's not always precise, and it can be time-consuming. Luckily, there's a formula we can use that makes this much easier. But before we get to the formula, let's solidify our understanding of what parallel lines mean in the context of their equations. Remember, the coefficients of x and y tell us about the slope (or gradient) of the line. If those coefficients are proportional, the lines have the same slope and are therefore parallel. This understanding will not only help us solve this problem but also tackle similar problems in the future. So, with our conceptual foundation in place, let's move on to the tools and techniques we'll need to actually calculate the distance!

The Formula for the Distance Between Parallel Lines

Alright, let's talk about the magic formula! There's a neat little equation that allows us to calculate the distance between two parallel lines without having to graph them or do any complicated geometry. The formula is derived from the more general formula for the distance between a point and a line, but it's been simplified specifically for the case of parallel lines. Here it is:

Distance = |C1 - C2| / √(A² + B²)

Now, let's break down what each of these symbols represents. Don't worry; it's not as intimidating as it looks! In this formula:

• |C1 - C2| represents the absolute value of the difference between the constant terms in the equations of the two lines. Absolute value just means we take the positive value of the result, so if we get a negative number inside the bars, we just drop the negative sign. This is important because distance can't be negative!
• A and B are the coefficients of x and y, respectively, in the equations of the lines. Remember, the equations of our lines are in the form Ax + By + C = 0. Since the lines are parallel, A and B will be the same for both equations.
• √(A² + B²) is the square root of the sum of the squares of A and B. This part of the formula comes from the Pythagorean theorem and helps us normalize the distance calculation.

So, why does this formula work? At its heart, it's finding the perpendicular distance between the lines. The numerator, |C1 - C2|, gives us a measure of the vertical separation between the lines, while the denominator, √(A² + B²), scales this separation to account for the slope of the lines. Think of it as adjusting for the angle at which the lines are tilted. If the lines are horizontal (A = 0), the denominator simplifies to B, and the formula essentially calculates the vertical difference in the y-intercepts. If the lines are vertical (B = 0), the denominator simplifies to A, and the formula calculates the horizontal difference in the x-intercepts. For lines that are neither horizontal nor vertical, the denominator provides the necessary correction to give us the true perpendicular distance.

Now that we understand the formula and where it comes from, we're ready to apply it to our specific problem. Let's identify the values of A, B, C1, and C2 from our line equations and plug them into the formula. This is where the rubber meets the road, and we'll see how this formula makes our calculation straightforward and precise.

Applying the Formula to Our Problem

Okay, guys, it's time to get our hands dirty and actually use the formula we just discussed. We have the equations of our lines:

• r: x + 3y - 10 = 0
• s: x + 3y - 6 = 0

Let's identify the values of A, B, C1, and C2. Remember, these correspond to the coefficients in the general form of a line equation, Ax + By + C = 0.

• A, the coefficient of x, is 1 in both equations.
• B, the coefficient of y, is 3 in both equations.
• C1, the constant term in the equation for line r, is -10.
• C2, the constant term in the equation for line s, is -6.

Now we have all the pieces we need! Let's plug these values into our formula for the distance between parallel lines:

Distance = |C1 - C2| / √(A² + B²)

Distance = |-10 - (-6)| / √(1² + 3²)

Time for some arithmetic! First, let's simplify the numerator:

|-10 - (-6)| = |-10 + 6| = |-4| = 4

Remember, the absolute value makes the result positive. Now, let's simplify the denominator:

√(1² + 3²) = √(1 + 9) = √10

So our equation now looks like this:

Distance = 4 / √10

We're almost there! It's generally good practice to rationalize the denominator, which means getting rid of the square root in the bottom of the fraction. To do this, we multiply both the numerator and the denominator by √10:

Distance = (4 / √10) * (√10 / √10) = 4√10 / 10

We can simplify this fraction further by dividing both the numerator and the denominator by 2:

Distance = 2√10 / 5

And that's it! We've calculated the distance between the lines r and s. The distance is 2√10 / 5 units. This is the exact distance, and it's the most mathematically precise way to express the answer. If you needed an approximate decimal value, you could plug √10 into a calculator (it's approximately 3.16) and then do the calculation.

So, we’ve successfully navigated the formula, plugged in our values, and arrived at our answer. But before we celebrate too much, let's take a step back and think about what we've done and how we might check our work. This kind of reflection is crucial for solidifying our understanding and making sure we really