Finding Original Coordinates After Transformation

Hey guys! Ever found yourself staring at a geometry problem involving transformations and feeling totally lost? Well, you're not alone! Today, we're going to break down a classic problem: finding the original coordinates of a point after it's been dilated and rotated. This might sound intimidating, but trust me, with a little step-by-step guidance, you'll be a pro in no time. So, grab your pencils, and let's dive in!

Understanding Transformations: Dilation and Rotation

Before we jump into solving the problem, let's quickly recap what dilation and rotation actually mean in the world of geometry. Think of it like this: transformations are simply ways to move and resize shapes on a coordinate plane. Each transformation has a specific rule that changes the position or size of the original shape, which we call the pre-image, to create a new shape, known as the image.

Dilation: Resizing the Shape

Dilation is all about scaling. It's like using a zoom lens on a camera – you can make the image bigger or smaller. The key here is the scale factor. A scale factor greater than 1 enlarges the shape, while a scale factor between 0 and 1 shrinks it. The center of dilation is the fixed point from which the shape is scaled. In our problem, we have a dilation represented by [0, 5]. This notation tells us two things: the center of dilation is at the origin (0, 0), and the scale factor is 5. This means that every point on the original shape will be five times further away from the origin in the dilated image.

Think of it like stretching a rubber band from the origin; the further you pull, the larger the shape gets. Dilation is a transformation that changes the size of a figure without changing its shape. The scale factor determines whether the figure is enlarged or reduced. When the scale factor is greater than 1, the figure is enlarged. When the scale factor is between 0 and 1, the figure is reduced. The center of dilation is the fixed point from which the dilation is performed. In this case, the dilation $[0, 5]$ indicates a scale factor of 5 with the origin (0, 0) as the center of dilation. This means every point on the original figure will be five times farther from the origin after the dilation.

Understanding the Scale Factor: The scale factor is crucial in dilation. It's the number you multiply the coordinates of the original point by to get the coordinates of the dilated point. If the scale factor is 2, the image will be twice as big as the original. If the scale factor is 0.5, the image will be half the size. Remember, dilation changes the size but not the shape.

Center of Dilation: The center of dilation is the fixed point around which the figure is enlarged or reduced. If the center of dilation is the origin (0,0), the calculations are straightforward. However, if the center of dilation is a different point, you'll need to adjust the coordinates relative to that center before applying the scale factor.

Rotation: Spinning the Shape

Rotation, on the other hand, is about turning a shape around a fixed point, called the center of rotation. We describe rotations by the angle of rotation (how much we turn it) and the direction (clockwise or counterclockwise). In our problem, we have a rotation of 180 degrees (180°) about the origin. A 180-degree rotation essentially flips the shape across both the x-axis and the y-axis. It's like spinning a wheel halfway around.

Imagine placing a shape on a turntable and spinning it; that's rotation in action! Rotation is a transformation that turns a figure about a fixed point. The center of rotation is the point around which the figure is rotated, and the angle of rotation specifies how much the figure is turned. A 180-degree rotation about the origin means the figure is turned halfway around, resulting in a point $(x, y)$ transforming to $(-x, -y)$. This is because a 180-degree rotation reflects the point across both the x-axis and the y-axis.

Understanding the Angle and Direction: The angle of rotation tells you how much the shape is turned. Common angles include 90 degrees, 180 degrees, and 270 degrees. The direction of rotation is either clockwise (CW) or counterclockwise (CCW). By convention, a positive angle indicates a counterclockwise rotation, while a negative angle indicates a clockwise rotation. A 180-degree rotation is the same whether it's clockwise or counterclockwise.

Center of Rotation: Just like dilation, the center of rotation is crucial. If the center of rotation is the origin, the transformation rules are simpler. However, if the center of rotation is a different point, the calculations become more complex, often involving translations to and from the center of rotation.

The Problem: Unraveling the Transformations

Okay, now that we've refreshed our understanding of dilation and rotation, let's tackle the problem at hand. We're given that point B ((x, y)) undergoes two transformations:

1. Dilation: Dilation by [0, 5] (center at the origin, scale factor of 5).
2. Rotation: Rotation of 180 degrees about the origin.

After these transformations, the image of B, called B'', ends up at (-10, -25). Our mission is to find the original coordinates of B.

Think of it as a detective story; we need to reverse the steps to find the original suspect (point B)! The key to solving this problem is working backward, reversing each transformation step by step. Start with the final image, B''(-10, -25), and undo the rotation first, then undo the dilation. This approach breaks the problem into manageable parts, making it easier to understand and solve.

Why Reverse the Order? The order in which transformations are applied matters. If we dilated first and then rotated, the final image would be different than if we rotated first and then dilated. Therefore, when reversing transformations, we must undo them in the reverse order they were applied. This ensures we accurately trace back to the original point.

Step-by-Step Solution: Working Backwards

To find the original coordinates of point B, we need to reverse the transformations in the opposite order they were applied. This means we'll first undo the rotation and then undo the dilation. Let's break it down:

1. Undoing the Rotation

The last transformation that happened was a 180-degree rotation about the origin. Remember, a 180-degree rotation changes the signs of both coordinates. So, if B'' is (-10, -25), to undo this rotation, we simply change the signs back.

It's like flipping a switch; we're just reversing the sign change caused by the rotation. To undo a 180-degree rotation, we apply the same transformation again because rotating a point 180 degrees twice brings it back to its original position. This principle simplifies the process of reversing the rotation.

Applying the Reverse Rotation: If a point (x, y) becomes (-x, -y) after a 180-degree rotation, then a point (-x, -y) will become (x, y) after another 180-degree rotation. This symmetry is key to reversing the transformation efficiently.

Therefore, before the rotation, the point (let's call it B') must have been (10, 25). So, B' is the point before the 180-degree rotation but after the dilation.

2. Undoing the Dilation

Now, we need to undo the dilation. We know the dilation was by a scale factor of 5 with the center at the origin. Dilation by a scale factor of 5 means the coordinates were multiplied by 5. To undo this, we need to divide the coordinates by 5.

Think of it as shrinking the shape back to its original size; we're doing the opposite of the zoom-in. Undoing the dilation involves applying the inverse scale factor. If the original dilation multiplied the coordinates by 5, the reverse operation divides the coordinates by 5.

Applying the Inverse Scale Factor: To reverse the effect of a dilation with a scale factor of 5, divide both coordinates of the dilated point by 5. This returns the point to its original position before the dilation was applied.

So, we take the coordinates of B', which are (10, 25), and divide each by 5:

• x-coordinate: 10 / 5 = 2
• y-coordinate: 25 / 5 = 5

Therefore, the original coordinates of point B are (2, 5).

The Answer: Cracking the Code

So, after reversing both the rotation and the dilation, we've found that the original coordinates of point B are (2, 5). This corresponds to option A in the multiple-choice answers provided.

We did it! We successfully traced the point back through the transformations to its origin. By systematically reversing each transformation, we were able to find the original coordinates of point B. This step-by-step approach is essential for solving complex transformation problems.

Reviewing the Steps: To recap, we first reversed the 180-degree rotation by changing the signs of the coordinates of B''. Then, we reversed the dilation by dividing the coordinates of the resulting point by the scale factor. This process highlights the importance of understanding the inverse operations of transformations.

Tips and Tricks for Transformation Problems

Before we wrap up, let's go over some helpful tips and tricks that can make solving transformation problems a whole lot easier:

• Visualize: Sketching the transformations can often help you understand what's happening and prevent errors. Even a rough sketch can provide valuable insights into the movement of the point or shape.
• Work Backwards: When you're given the final image and need to find the original, always work backwards, reversing the transformations in the opposite order they were applied. This strategy breaks the problem into smaller, more manageable steps.
• Know the Rules: Memorize the transformation rules for common transformations like rotations (90, 180, 270 degrees), reflections, and dilations. Knowing these rules by heart will speed up your problem-solving process and reduce the chance of making mistakes.
• Practice Makes Perfect: The more you practice these types of problems, the more comfortable you'll become with them. Work through various examples, and don't be afraid to try different approaches until you find what works best for you.

Mastering transformations takes time and practice, but the effort is well worth it! By consistently applying these tips and tricks, you'll develop a strong understanding of transformations and be able to tackle even the most challenging problems with confidence.

Wrapping Up: You've Got This!

So, there you have it! We've successfully navigated the world of dilations and rotations to find the original coordinates of a point. Remember, the key is to understand the transformations, work backwards, and take it one step at a time. With practice, you'll be transforming like a pro in no time!

Keep practicing, keep exploring, and remember that every problem is an opportunity to learn something new! Transformations are a fundamental concept in geometry, and mastering them will not only help you in math class but also in various real-world applications, such as computer graphics and engineering.

I hope this breakdown has been helpful. Keep up the great work, and I'll catch you in the next math adventure! Remember, math can be fun when you approach it with the right mindset and a willingness to learn. Keep challenging yourself, and you'll be amazed at what you can achieve!