Dilations And Translations: Finding The Image Of Point B
Hey guys! Let's dive into a super interesting problem involving dilations and translations in geometry. We've got a point A that undergoes a couple of transformations, and we need to use that information to figure out what happens to another point, B, when it goes through the same process. It sounds like a puzzle, right? But trust me, it's super fun once you get the hang of it. So, grab your pencils and let's get started!
Understanding the Transformations
Before we jump into the actual problem, let's quickly recap what dilations and translations are. Dilations are transformations that change the size of a figure, either making it bigger (enlargement) or smaller (reduction). This change in size is determined by a scale factor, which we often call 'k'. If 'k' is greater than 1, the figure gets bigger; if 'k' is between 0 and 1, it gets smaller. The center of dilation is the fixed point from which the figure expands or contracts. In our case, the center of dilation is the origin, O(0, 0), which makes things a bit simpler.
Translations, on the other hand, are transformations that slide a figure from one place to another without changing its size or orientation. We simply move the figure a certain number of units horizontally and vertically. This is often described as a “shift.” Think of it like sliding a piece of paper across a table – the paper itself doesn't change, just its position. So, now that we have a solid grasp of what dilations and translations mean, let's circle back to our specific problem. We know point A undergoes dilation followed by translation. This two-step process is key to understanding how point B will transform as well. We're essentially reverse-engineering the process using the information from point A to apply it to point B.
The beauty of geometry, guys, is that it's all about relationships and patterns. If we can understand how one point behaves under certain transformations, we can apply that same understanding to other points as well. It's like cracking a code! And that's exactly what we're going to do here. Let's break down the transformations step by step to make sure we're crystal clear on what's happening. By carefully analyzing the movements of point A, we're setting ourselves up for success in figuring out what happens to point B. Remember, the key is to take it one step at a time and not get overwhelmed by the whole process. We've got this!
Step 1: Decoding the Dilation
Okay, let's start by focusing on the dilation part of the transformation. We know that point A(1, -2) is dilated with a scale factor 'k' and the center of dilation is O(0, 0). The result of this dilation is an intermediate point, let's call it A1. We don't know the coordinates of A1 yet, but we know that the dilation changes the distance of A from the origin by a factor of 'k'. So, the coordinates of A1 will be (k * 1, k * -2), which simplifies to (k, -2k). This is because each coordinate of the original point is multiplied by the scale factor 'k'. Think of it like scaling up or down a blueprint – everything gets proportionally larger or smaller. We are using the origin as the anchor point, so the dilation is simply a multiplication of coordinates. If the center of dilation were somewhere else, things would get a little more complex, but for now, we're keeping it nice and straightforward.
So, after the dilation, point A has moved from (1, -2) to (k, -2k). The value of 'k' is super important here because it determines how much bigger or smaller the figure gets. If 'k' is 2, the figure doubles in size; if 'k' is 0.5, it halves in size. In our case, we don't know 'k' yet, but we're going to figure it out using the information about the translation that comes next. This is where the puzzle pieces start to fit together! We're using the properties of dilation to set up an equation that will eventually help us find the value of 'k'. It's all about using what we know to find what we don't know. This is a fundamental principle in mathematics, and it's what makes problem-solving so satisfying. We're taking a seemingly complex problem and breaking it down into smaller, more manageable steps. And that, my friends, is the key to success!
Step 2: Unraveling the Translation
Now, let's shift our focus to the translation part of the transformation. We know that after the dilation, the intermediate point A1(k, -2k) is translated 3 units to the right and 2 units down. This means we're adding 3 to the x-coordinate and subtracting 2 from the y-coordinate. Remember, translations are all about shifting positions without changing size or shape. It's like moving a chess piece across the board – the piece itself stays the same, but its location changes.
So, after the translation, the final image of point A, which we're calling A', will have coordinates (k + 3, -2k - 2). We know that A' is given as (5, -6). This is the crucial piece of information that will allow us to solve for 'k'. We've now expressed the final coordinates of A' in terms of 'k', and we also know the actual coordinates of A'. This sets up a system of equations that we can solve. Think of it like detective work – we have a clue (the coordinates of A') and we're using it to track down the missing piece of the puzzle ('k'). The beauty of this approach is that it breaks down a complex problem into smaller, more manageable parts. We've tackled the dilation, we've tackled the translation, and now we're ready to put it all together to find the solution. It's like building a house – you lay the foundation, then the walls, then the roof. Each step builds on the previous one, and eventually, you have a complete structure.
Solving for the Scale Factor (k)
Alright, guys, this is where the magic happens! We've got the coordinates of A' in terms of 'k' (k + 3, -2k - 2) and we know A' is actually (5, -6). This means we can set up two equations:
- k + 3 = 5
- -2k - 2 = -6
Let's solve the first equation for 'k'. Subtracting 3 from both sides, we get:
k = 5 - 3 k = 2
Now, just to be sure, let's check if this value of 'k' works in the second equation. Substituting k = 2 into the second equation, we get:
-2(2) - 2 = -6 -4 - 2 = -6 -6 = -6
It checks out! So, we've successfully found the scale factor, k = 2. This is a major breakthrough because now we know exactly how much the dilation is scaling the figure. It's like finding the key to a locked door – once you have the key, you can unlock all sorts of possibilities. In this case, knowing 'k' allows us to apply the same transformations to point B and find its final image. We're building on our knowledge, step by step, and that's what makes math so powerful. We're not just memorizing formulas; we're understanding the underlying principles and using them to solve problems. And that's a skill that will take you far in life, not just in math class!
Applying the Transformations to Point B
Okay, now that we know the scale factor k = 2, we can finally figure out what happens to point B(0, 3) when it undergoes the same transformations. Remember, the transformations are dilation with a scale factor of 2, followed by a translation of 3 units to the right and 2 units down. Let's take it one step at a time, just like we did with point A. This methodical approach is key to avoiding mistakes and ensuring we get the correct answer. Think of it like following a recipe – if you follow the steps carefully, you're much more likely to end up with a delicious dish. And in this case, our delicious dish is the final coordinates of point B!
Step 1: Dilating Point B
First, we dilate point B(0, 3) with a scale factor of 2. This means we multiply both the x-coordinate and the y-coordinate by 2. So, the intermediate point, let's call it B1, will have coordinates (2 * 0, 2 * 3), which simplifies to (0, 6). Notice how the dilation has changed the position of the point relative to the origin. It's like zooming in on a map – the relative positions of landmarks change as you zoom in or out. In this case, the dilation is stretching point B away from the origin by a factor of 2. This is a visual way to understand what's happening, and it can help you remember the concept of dilation. It's not just about multiplying numbers; it's about changing the size and position of a figure in a specific way.
Step 2: Translating Point B1
Next, we translate point B1(0, 6) by 3 units to the right and 2 units down. This means we add 3 to the x-coordinate and subtract 2 from the y-coordinate. So, the final image of point B, which we'll call B', will have coordinates (0 + 3, 6 - 2), which simplifies to (3, 4). And there you have it! We've successfully found the final image of point B after undergoing the same transformations as point A. It's like completing a journey – we started with point B, we applied the transformations, and we arrived at our destination, B'. Each step of the journey was important, and by carefully following the steps, we made sure we didn't get lost along the way.
The Final Image: B'(3, 4)
So, after all that awesome work, we've found that the final image of point B, after being dilated and translated, is B'(3, 4). How cool is that? We took a problem that seemed a bit complex at first, and we broke it down into manageable steps. We used the information about point A to figure out the scale factor, and then we applied that knowledge to point B. This is a classic example of how math can be used to solve real-world problems. Whether you're designing a building, creating a video game, or even just planning a trip, the principles of geometry and transformations can be incredibly useful.
And that's the beauty of mathematics, guys. It's not just about memorizing formulas and procedures; it's about understanding the underlying concepts and applying them to solve problems. It's about thinking critically, breaking down complex situations, and finding creative solutions. So, the next time you encounter a problem that seems daunting, remember the steps we took here. Break it down, focus on one step at a time, and don't be afraid to ask for help. You've got this! Keep practicing, keep exploring, and keep having fun with math!
I hope you enjoyed this explanation! Let me know if you have any other questions or want to tackle another problem together. Keep rocking those math skills!