Distance From Point (6,4) To Line Y=x+4: Calculation Guide

Hey guys! Today, we're diving into a fundamental concept in coordinate geometry: finding the distance from a point to a line. Specifically, we're going to tackle the problem of calculating the distance from the point (6,4) to the line y = x + 4. This is a classic problem that combines algebraic manipulation with geometric understanding. So, buckle up, and let's get started!

Understanding the Core Concept

Before we jump into the specifics, let's make sure we understand the underlying concept. When we talk about the distance from a point to a line, we're referring to the shortest distance. This shortest distance is always the perpendicular distance – that is, the length of the line segment drawn from the point to the line such that it forms a right angle with the line. Thinking about this geometrically is crucial, as it guides our approach to solving the problem. We're not just looking for any distance; we're looking for the one that forms that perfect 90-degree angle. This is where the concept of perpendicularity plays a major role in the calculation, and understanding this will make the process much clearer.

Now, you might be wondering, why perpendicular distance? Well, imagine drawing various lines from the point to the given line. You'll notice that the shortest one is always the one that hits the line at a right angle. Any other line would be longer, forming the hypotenuse of a right triangle. This visual understanding is key to grasping the concept. It's not just about applying a formula; it's about understanding why the formula works.

To find this perpendicular distance, we typically use a specific formula derived from the general equation of a line and the coordinates of the point. This formula might seem a bit intimidating at first, but don't worry, we'll break it down step-by-step. The key is to understand where the formula comes from and how it relates to the geometry of the problem. Once you've got that down, you'll be able to tackle similar problems with confidence. Remember, math is all about building a strong foundation of understanding, and that's what we're aiming for here.

The Formula for Point-to-Line Distance

The formula to calculate the distance (d) from a point (x₁, y₁) to a line given by the equation Ax + By + C = 0 is:

d = |Ax₁ + By₁ + C| / √(A² + B²)

This formula might look a bit daunting at first, but let's break it down. Each term in the formula has a specific purpose, and understanding what they represent will make the formula much less intimidating. Think of it as a toolbox – each part has a job to do.

• The numerator, |Ax₁ + By₁ + C|, involves substituting the coordinates of the point (x₁, y₁) into the left-hand side of the line's equation (after it's been rearranged into the form Ax + By + C = 0). The absolute value ensures that the distance is always positive, which makes sense because distance is a measure of length, and length can't be negative. This part essentially calculates a value related to how far the point is from satisfying the equation of the line.
• The denominator, √(A² + B²), involves the coefficients A and B from the line's equation. These coefficients are related to the slope and orientation of the line. Squaring them, adding them, and then taking the square root gives us a normalizing factor. This normalizing factor ensures that the distance we calculate is the perpendicular distance, not just any distance.

So, in essence, the formula is a way of measuring how much the point 'misses' satisfying the line's equation, and then scaling that 'miss' by a factor that accounts for the line's orientation. It’s a clever combination of algebra and geometry that gives us the precise perpendicular distance we're looking for. Remember, the key to mastering this formula is practice and understanding the logic behind it. Don't just memorize it; understand it!

Applying the Formula to Our Problem

Now, let's apply this formula to our specific problem: finding the distance from the point (6,4) to the line y = x + 4. The first thing we need to do is rewrite the equation of the line in the standard form, Ax + By + C = 0. To do this, we subtract y from both sides to get:

x - y + 4 = 0

Now we can identify the coefficients: A = 1, B = -1, and C = 4. We also have our point (x₁, y₁) = (6, 4). We've got all the pieces we need to plug into the formula! This is a crucial step, so make sure you take your time and get it right. Transforming the equation into the standard form is essential for correctly identifying A, B, and C, and any mistake here will throw off your final answer. Think of it as setting the stage for the main event – you need everything in place before you can proceed.

Now, let's substitute these values into the distance formula:

d = |(1)(6) + (-1)(4) + 4| / √(1² + (-1)²)

We're simply replacing the variables in the formula with the numbers we've identified. This is where careful arithmetic comes into play. Double-check your substitutions to make sure you haven't mixed anything up. It's easy to make a small mistake at this stage, but it's also easy to avoid by being meticulous. Treat it like a puzzle – each number has its place, and the final picture will only be correct if everything fits together perfectly.

Next, we'll simplify the expression. This involves performing the arithmetic operations in the correct order (PEMDAS/BODMAS, remember?). We'll multiply, add, subtract, and finally take the square root. This is where our algebraic skills come to the forefront. It's not just about following the formula; it's about executing the calculations accurately. Stay focused, take it one step at a time, and you'll be well on your way to finding the solution.

Step-by-Step Calculation

Let's walk through the calculation step-by-step to make sure we don't miss anything. We've already substituted the values into the formula, so we have:

d = |(1)(6) + (-1)(4) + 4| / √(1² + (-1)²)

First, we perform the multiplications inside the absolute value:

d = |6 - 4 + 4| / √(1² + (-1)²)

Next, we add and subtract inside the absolute value:

d = |6| / √(1² + (-1)²)

Now, we calculate the squares in the denominator:

d = 6 / √(1 + 1)

Then, we add the numbers under the square root:

d = 6 / √2

To rationalize the denominator (which is a standard practice in mathematics), we multiply both the numerator and the denominator by √2:

d = (6√2) / (√2 * √2) = (6√2) / 2

Finally, we simplify the fraction:

d = 3√2

So, the exact distance is 3√2 units. But the question asks us to round to the nearest tenth, so we need to approximate the value of √2 (which is approximately 1.414) and then multiply by 3:

d ≈ 3 * 1.414 ≈ 4.242

Rounding to the nearest tenth, we get:

d ≈ 4.2

Therefore, the distance from the point (6,4) to the line y = x + 4 is approximately 4.2 units. Remember, each step is important, and accuracy is key. From substituting the values to simplifying the expression, every operation contributes to the final result. By breaking down the problem into manageable steps, we can ensure that we arrive at the correct answer with confidence.

Conclusion

So there you have it, guys! We've successfully calculated the distance from the point (6,4) to the line y = x + 4, rounding to the nearest tenth. We started by understanding the concept of perpendicular distance, then introduced the distance formula, and finally applied the formula step-by-step to solve the problem. Remember, the key to mastering these types of problems is practice and understanding the underlying principles. Don't just memorize formulas; understand where they come from and why they work.

This problem is a great example of how algebra and geometry come together to solve real-world problems. The distance formula isn't just an abstract mathematical concept; it's a tool we can use to measure distances in a coordinate plane. And that's pretty cool, right? By understanding this formula and how to apply it, you've added another valuable tool to your mathematical toolkit. So keep practicing, keep exploring, and keep challenging yourselves. The world of mathematics is full of fascinating concepts and problems waiting to be discovered!

If you have any questions or want to explore more problems like this, feel free to ask! Keep up the great work, and I'll see you in the next math adventure!