Finding X For Equidistant Point A From B And C A Comprehensive Guide

Hey guys! Let's dive into an exciting math problem today where we need to find the x-coordinate of a point A such that it's equidistant from two other points, B and C. This kind of problem pops up quite often in coordinate geometry, and it’s super useful for understanding distances and geometric relationships on a plane. We'll break it down step by step, so you can totally nail it! So, grab your pencils, and let’s get started!

Problem Statement

Okay, so here’s the deal. We've got three points: A, B, and C. Point A has coordinates (x, 5), which means we know its y-coordinate is 5, but we need to figure out its x-coordinate. Point B is located at (-2, 3), and point C is at (4, 1). The key here is that point A is equidistant from both B and C. This means the distance from A to B is exactly the same as the distance from A to C. Our mission, should we choose to accept it (and we totally do!), is to find the value of x that makes this true. This involves using the distance formula, setting up an equation, and solving for x. Remember, the distance formula is our best friend in these scenarios because it tells us exactly how to calculate the distance between two points in a coordinate plane. Understanding and applying this formula is crucial for tackling this kind of problem effectively. We'll walk through each step so you can see exactly how it’s done, and by the end, you’ll be a pro at solving these equidistant point problems. Let's jump in and see how it all works!

Understanding the Distance Formula

Alright, before we jump into the calculations, let's make sure we're all comfy with the distance formula. This formula is the backbone of solving problems involving distances between points in a coordinate plane. It's derived straight from the Pythagorean theorem (remember a² + b² = c²?), so if you're familiar with that, you're already halfway there! The distance formula tells us how to find the distance between any two points (x₁, y₁) and (x₂, y₂). It looks a little something like this:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

So, what’s really going on here? Let's break it down. The formula calculates the distance as the square root of the sum of the squares of the differences in the x-coordinates and the y-coordinates. Basically, we’re finding the lengths of the horizontal and vertical sides of a right triangle and then using the Pythagorean theorem to find the hypotenuse, which is the distance between our two points. Now, why is this important? Well, in our problem, we need to find a point A that is equidistant from B and C. This means the distance from A to B must be equal to the distance from A to C. To figure that out, we need to calculate both distances using this formula. Understanding this formula intuitively is key, so take a moment to really digest it. Once you've got this down, applying it to our specific problem becomes a whole lot easier. We'll use it to set up an equation that we can then solve for our unknown x-coordinate. So, let’s make sure we’re all on the same page with this, and then we’ll move on to the exciting part: plugging in our coordinates and getting our hands dirty with some math!

Applying the Distance Formula to Points A and B

Okay, now that we've got the distance formula fresh in our minds, let's apply it to the specific points we're dealing with: A(x, 5) and B(-2, 3). We want to find the distance between these two points, so we'll plug their coordinates into our trusty formula. Remember, the distance formula is √[(x₂ - x₁)² + (y₂ - y₁)²]. Let's call A's coordinates (x₁, y₁) and B's coordinates (x₂, y₂). So, x₁ is x, y₁ is 5, x₂ is -2, and y₂ is 3. Now, let’s plug these values into the formula and see what we get. The distance AB can be calculated as follows:

AB = √[(-2 - x)² + (3 - 5)²]

See how we just replaced the x's and y's in the formula with our specific coordinates? That’s the first step, and it's super important to get it right. Now, let's simplify this a bit. We've got (-2 - x)² and (3 - 5)². The (3 - 5)² part is easy: that's just (-2)², which equals 4. The (-2 - x)² part is a little trickier because we need to square a binomial. Remember, (a + b)² is a² + 2ab + b², so we'll use that to expand (-2 - x)². This step is crucial because simplifying the expression inside the square root will make our lives much easier when we start setting up equations. Take your time with this part, double-check your signs, and make sure you're comfortable with the expansion. Once we’ve simplified this expression, we’ll have a clear representation of the distance AB in terms of x. This is a critical step in solving our problem because it allows us to relate this distance to the distance AC, which we'll calculate next. So, let's keep going, simplify this expression, and get one step closer to finding the value of x!

Calculating the Distance Between Points A and C

Alright, let's keep the momentum going! Now that we've found the distance between points A and B in terms of x, it's time to do the same for the distance between points A(x, 5) and C(4, 1). We're going to use the same distance formula again, which is √[(x₂ - x₁)² + (y₂ - y₁)²]. This time, let's consider A as (x₁, y₁) and C as (x₂, y₂). So, x₁ is still x, y₁ is still 5, but now x₂ is 4 and y₂ is 1. Plugging these values into the formula, we get:

AC = √[(4 - x)² + (1 - 5)²]

Just like before, we've substituted the coordinates into the formula. Now, our next step is to simplify this expression. We've got (4 - x)² and (1 - 5)². Let's start with the easy part: (1 - 5)² is (-4)², which equals 16. Now for the slightly trickier part: (4 - x)². Again, we're squaring a binomial, so we need to remember our formula (a - b)² = a² - 2ab + b². Expanding (4 - x)² carefully is super important because any mistake here will throw off our final answer. We need to make sure we get all the signs right and combine like terms correctly. Simplifying this expression is another key step in our problem-solving process. Once we have a simplified expression for AC, we'll have all the pieces we need to set up our equation. We know that AB and AC must be equal since point A is equidistant from B and C. So, by calculating both distances and simplifying their expressions, we're setting ourselves up to solve for x. Let's dive into simplifying this expression, and then we'll be ready to equate the two distances and find our x-coordinate!

Setting Up the Equation AB = AC

Okay, fantastic! We've calculated the distances AB and AC, and now we're at a crucial point: setting up the equation. Remember, the problem states that point A is equidistant from points B and C. What does that mean for us? It means the distance AB is exactly the same as the distance AC. So, we can confidently write this as an equation:

AB = AC

Now, let’s substitute the expressions we found earlier for AB and AC into this equation. This is where all our hard work pays off! We'll have an equation with square roots and some algebraic terms, and our goal is to solve for x. The equation will look something like this:

√[(-2 - x)² + (3 - 5)²] = √[(4 - x)² + (1 - 5)²]

See how we've just replaced AB and AC with their respective expressions? This is a significant step because it translates the geometric condition (equidistant) into an algebraic equation that we can actually solve. This equation might look a little intimidating at first glance, with those square roots and binomials, but don't worry, we've got a plan to tackle it. The next step is to get rid of those pesky square roots so we can work with a simpler equation. Remember, squaring both sides of an equation is a valid operation as long as we do it to the entire side. This will eliminate the square roots and give us a more manageable equation to solve. Setting up this equation correctly is absolutely key to finding the right value for x. If we make a mistake here, our final answer will be off. So, double-check that you've correctly substituted the expressions for AB and AC, and then let's move on to the next step: squaring both sides and simplifying!

Solving for x

Alright, let's get down to the nitty-gritty and solve for x! We've set up our equation, which looks something like √[(-2 - x)² + (3 - 5)²] = √[(4 - x)² + (1 - 5)²]. The first thing we want to do is get rid of those square roots. How do we do that? Easy peasy – we square both sides of the equation! Squaring both sides gives us:

[(-2 - x)² + (3 - 5)²] = [(4 - x)² + (1 - 5)²]

Now, we've got a cleaner equation to work with. Let's simplify each side. Remember, we already know that (3 - 5)² is (-2)² which equals 4, and (1 - 5)² is (-4)² which equals 16. So, our equation becomes:

[(-2 - x)² + 4] = [(4 - x)² + 16]

Next up, we need to expand those squared binomials. Recall that (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². So, let's expand (-2 - x)² and (4 - x)²:

(-2 - x)² = 4 + 4x + x² (4 - x)² = 16 - 8x + x²

Now, let's substitute these back into our equation:

[4 + 4x + x² + 4] = [16 - 8x + x² + 16]

Time to simplify further by combining like terms:

x² + 4x + 8 = x² - 8x + 32

Notice that we have x² on both sides of the equation. This is great news because we can subtract x² from both sides, and it'll cancel out, leaving us with a linear equation. So, let's do that:

4x + 8 = -8x + 32

Now, it's just a matter of isolating x. Let's add 8x to both sides:

12x + 8 = 32

Subtract 8 from both sides:

12x = 24

And finally, divide both sides by 12:

x = 2

Boom! We found it! The x-coordinate of point A is 2. That means point A is located at (2, 5). This was a fantastic journey, and we solved it step by step. Make sure you go through each step carefully to really understand the process. Next, we'll quickly verify our solution to make sure everything checks out!

Verifying the Solution

Okay, now that we've found x = 2, the smart thing to do is verify our solution. We want to make sure that point A(2, 5) is indeed equidistant from B(-2, 3) and C(4, 1). To do this, we'll calculate the distances AB and AC using our newfound x-coordinate and see if they're the same. First, let's calculate the distance AB:

AB = √[(-2 - 2)² + (3 - 5)²] = √[(-4)² + (-2)²] = √(16 + 4) = √20

Now, let's calculate the distance AC:

AC = √[(4 - 2)² + (1 - 5)²] = √[(2)² + (-4)²] = √(4 + 16) = √20

Look at that! AB and AC both equal √20. This confirms that our solution x = 2 is correct! Point A(2, 5) is indeed equidistant from points B and C. Verifying our solution is super important because it helps us catch any mistakes we might have made along the way. It’s like the final piece of the puzzle that makes everything click into place. By plugging our value of x back into the original problem, we can be confident that we've solved it correctly. This step not only gives us peace of mind but also reinforces our understanding of the problem and the steps we took to solve it. So, always remember to verify your solution whenever you can. It's a best practice that will save you headaches in the long run!

Conclusion

Alright, guys, we did it! We successfully found the x-coordinate for point A such that A is equidistant from points B and C. We started by understanding the problem, then we dusted off our trusty distance formula, applied it to points A and B, and then to points A and C. We set up an equation equating the distances AB and AC, solved for x, and even verified our solution to make sure we nailed it. This kind of problem is a classic example of how coordinate geometry combines algebra and geometry to solve real problems. It’s not just about memorizing formulas; it’s about understanding how those formulas represent geometric relationships and using them to find unknown values. The key takeaways from this problem are the importance of the distance formula, the process of setting up equations based on geometric conditions, and the algebraic techniques for solving those equations. But perhaps the most important takeaway is the value of verifying your solution. It’s a simple step that can save you from making mistakes and ensure that you truly understand the problem. So, the next time you encounter a similar problem, remember the steps we followed today, and you’ll be well on your way to solving it with confidence. Great job, and keep up the awesome work!