Rotating Polygon PSRT 270 Degrees Counterclockwise A Step-by-Step Solution

Hey guys! Today, we're diving into the fascinating world of geometric transformations, specifically focusing on rotations. Our mission? To figure out what happens when we spin a polygon named PSRT by a whopping 270 degrees counterclockwise. So, buckle up and let's get started!

The Polygon PSRT and Its Vertices

First things first, let's get acquainted with our polygon. We have Polygon PSRT, and it's defined by its vertices (those corner points, you know?). These vertices are:

• P (1, 5)
• S (1, 0)
• R (-2, -3)
• T (-4, 2)

Think of these as the addresses of our polygon's corners on a coordinate plane. Now, our goal is to find the new address, the image vertices, of point R (which we'll call R') after we rotate the entire polygon 270 degrees counterclockwise. This involves understanding the rules of rotation and how they affect coordinates.

The Magic of Rotations in Mathematics

In the realm of mathematics, rotations are a fundamental type of transformation. Understanding rotations is crucial in various fields, from computer graphics to physics. A rotation involves spinning a figure around a fixed point, known as the center of rotation. In our case, we're dealing with a 270-degree counterclockwise rotation. Let's break down what that means and how it affects the coordinates of a point. When rotating a point in a 2D plane, the coordinates change according to specific rules derived from trigonometric principles. These rules differ based on the angle of rotation and the direction (clockwise or counterclockwise). For a 270-degree counterclockwise rotation, the transformation can be described mathematically. Imagine the coordinate plane as a canvas, and we're taking the polygon PSRT and giving it a spin around the origin (the point (0,0)). A 270-degree counterclockwise rotation is the same as a 90-degree clockwise rotation. This equivalence is helpful because it provides us with another way to visualize and compute the transformation. The key to performing rotations effectively is understanding the underlying formulas and applying them meticulously. When we rotate a point 270 degrees counterclockwise around the origin, we're essentially moving it three-quarters of the way around a circle in the counterclockwise direction. This extensive rotation significantly alters the coordinates of the point, and the formula helps us track exactly how. Now, let's delve into the specific rules for this type of rotation so we can find the new coordinates of point R after the transformation.

Understanding 270° Counterclockwise Rotation

A 270° counterclockwise rotation might sound intimidating, but it's actually quite manageable once we understand the rules. Basically, it's like turning something three-quarters of a full circle in the opposite direction that clocks go. In the coordinate plane, rotating a point (x, y) 270 degrees counterclockwise around the origin (0, 0) results in a new point with coordinates (y, -x). This rule is derived from the geometric properties of rotation and the trigonometric relationships between angles and coordinates. To apply this rule effectively, you need to understand the relationship between the original coordinates and the transformed coordinates. Let's break it down: The original x-coordinate becomes the new y-coordinate, and the original y-coordinate becomes the negative of the new x-coordinate. This switch and sign change are crucial to correctly performing the rotation. Visualizing the rotation on the coordinate plane can also help solidify this concept. Imagine a point moving along a circular path around the origin. A 270-degree counterclockwise rotation means the point travels three-quarters of the circle in the counterclockwise direction. This visual representation can make the coordinate transformation rule more intuitive. The coordinate transformation rule is derived from trigonometric principles. When we rotate a point around the origin, its coordinates change according to sine and cosine functions. Specifically, a 270-degree counterclockwise rotation can be represented by the following transformations: x' = y and y' = -x, where (x, y) are the original coordinates and (x', y') are the new coordinates after rotation. This rule is essential for accurately determining the new position of any point after a 270-degree counterclockwise rotation. Now that we understand the rule, let's apply it to our specific point, R.

The Rule of Transformation: (x, y) becomes (y, -x)

So, here's the key: when we rotate a point (x, y) 270 degrees counterclockwise, it transforms into a new point (y, -x). Notice the switcheroo? The original y-coordinate becomes the new x-coordinate, and the original x-coordinate becomes the negative of the new y-coordinate. This nifty rule is what we'll use to find R'. This rule stems from the fundamental principles of coordinate geometry and how rotations affect the positions of points in the plane. To fully grasp this, it's helpful to visualize the coordinate plane and how the axes interchange during a 270-degree counterclockwise rotation. Imagine rotating the entire plane; the original x-axis becomes the new y-axis, and the original y-axis becomes the negative of the new x-axis. This physical rotation helps explain the coordinate transformation. Let's take a closer look at why this rule works. Consider a point in the first quadrant (where both x and y are positive). After a 270-degree counterclockwise rotation, this point will end up in the fourth quadrant, where the x-coordinate is positive and the y-coordinate is negative. This aligns with the transformation rule: the original y (positive) becomes the new x (positive), and the original x (positive) becomes the negative of the new y (negative). This logic applies to points in other quadrants as well, making the rule universally applicable for 270-degree counterclockwise rotations. The coordinate transformation rule is not just a mathematical formula; it's a powerful tool for understanding geometric transformations. Once you master this rule, you can easily predict how any point will move after a 270-degree counterclockwise rotation. Now, let's put this knowledge to practical use by applying the rule to point R and finding its new coordinates.

Finding R': Applying the Rule to Point R (-2, -3)

Alright, let's get down to business! We know point R has coordinates (-2, -3). We're going to use our trusty rule – (x, y) becomes (y, -x) – to find the coordinates of R'. So, we take our original point R (-2, -3) and apply the rule. First, we swap the x and y coordinates. The -3 becomes our new x-coordinate, and the -2 will be used to find our new y-coordinate. Next, we take the original x-coordinate (-2) and make it negative, giving us -(-2), which simplifies to 2. Therefore, after applying the rule, we find that the new coordinates of R', the image of R after a 270-degree counterclockwise rotation, are ( -3, 2). This process is straightforward but requires careful attention to detail, especially when dealing with negative numbers. It's helpful to double-check each step to ensure accuracy. Now, let's walk through the application of the rule one more time, emphasizing the importance of correctly handling the signs. We start with R (-2, -3). To find R', we first swap the coordinates, which gives us (-3, -2). Then, we take the original x-coordinate, -2, and make it negative, resulting in -(-2) = 2. This final step gives us the coordinates of R' as (-3, 2). Practicing these transformations with different points can help solidify your understanding and build confidence in your ability to apply the rule correctly. So, remember, swap the coordinates and negate the original x-coordinate. This simple yet powerful rule will allow you to accurately perform 270-degree counterclockwise rotations. Now that we've found the image vertex R', let's highlight the final answer and consider how this rotation affects the overall shape of the polygon.

The Calculation: R' = (-3, 2)

Following the rule, if R is (-2, -3), then R' becomes (-3, 2). We swapped the coordinates and made the original x-coordinate (-2) the negative of the new y-coordinate, which gives us 2. There you have it! The image vertex R' after the 270° counterclockwise rotation is (-3, 2). This result demonstrates the practical application of the coordinate transformation rule and provides a clear answer to our initial question. It's important to note that the transformation rule works consistently across all points, regardless of their initial quadrant in the coordinate plane. This consistency is a key characteristic of geometric rotations and allows us to accurately predict the new position of any point after rotation. To ensure our answer is correct, it's always a good idea to visualize the rotation on a coordinate plane. If you plot the original point R (-2, -3) and then imagine rotating it 270 degrees counterclockwise, you'll see that the new position (-3, 2) aligns perfectly with the rotation. This visual check can help prevent errors and reinforce your understanding of the transformation. Now that we've successfully found the image vertex R', let's take a moment to appreciate the power of these coordinate transformation rules and how they enable us to perform geometric operations precisely. Understanding and applying these rules is fundamental to more advanced topics in mathematics and computer graphics. So, congratulations! You've successfully navigated a 270-degree counterclockwise rotation and found the new coordinates of R'.

Visualizing the Rotation (Optional)

If you're a visual learner (like me!), it can be super helpful to actually see the rotation. Imagine the coordinate plane with the polygon PSRT plotted on it. Now, picture grabbing the polygon and spinning it 270 degrees counterclockwise around the origin. You'd see point R moving to its new location at (-3, 2). Tools like graphing software or even a simple hand-drawn sketch can make this visualization even clearer. Visualizing the rotation is a powerful way to confirm your calculations and deepen your understanding of geometric transformations. By seeing how the polygon moves and how its vertices change position, you can develop a more intuitive grasp of the underlying principles. For instance, you can observe how the distances between the vertices remain constant during the rotation, even though their coordinates change. This preservation of distances is a key characteristic of rotations, distinguishing them from other types of transformations like scaling or shearing. Visualizing the rotation also helps you understand the relationship between the original point and its image. You can see how a 270-degree counterclockwise rotation moves a point three-quarters of the way around a circle centered at the origin. This visual connection between the rotation angle and the point's movement can make the transformation rule (x, y) → (y, -x) more meaningful. Furthermore, visualizing the rotation can help you catch errors in your calculations. If your calculated image point doesn't align with what you see in the rotated image, it's a clear indication that you need to revisit your steps. So, whether you use graphing software, a hand-drawn sketch, or simply your imagination, visualizing geometric transformations is a valuable tool for learning and problem-solving. Now that we've visualized the rotation and found the image vertex R', let's recap the key steps and insights from this exploration.

Wrapping Up: Key Takeaways

So, we've successfully navigated the world of rotations and found the image vertex R' after a 270° counterclockwise spin. The key takeaway here is the rule: (x, y) transforms into (y, -x). Remember this, and you'll be a rotation whiz in no time! We've also seen how visualizing these transformations can make them easier to understand. Geometric transformations, such as rotations, are fundamental concepts in mathematics with wide-ranging applications. They play a crucial role in fields like computer graphics, where they are used to manipulate objects in 3D space, and in physics, where they describe the motion of objects. Understanding rotations also builds a foundation for learning more advanced topics in geometry and linear algebra. The coordinate transformation rule we used, (x, y) → (y, -x) for a 270-degree counterclockwise rotation, is just one example of many such rules for different transformations. By mastering these rules, you gain the ability to predict and control how geometric figures change under various transformations. Moreover, this exploration has highlighted the importance of meticulousness and attention to detail in mathematical problem-solving. Correctly applying the transformation rule and handling the signs of coordinates are essential for accurate results. Finally, remember that visualization is a powerful tool for understanding geometric concepts. Whether you use graphing software or simply imagine the transformation in your mind, visualizing rotations can deepen your understanding and improve your problem-solving skills. So, armed with this knowledge and practice, you're well-equipped to tackle more challenging problems involving geometric transformations. Congratulations on mastering 270-degree counterclockwise rotations! And remember, math isn't just about numbers and formulas; it's about understanding the world around us in a precise and logical way.

I hope this breakdown was helpful and made the rotation rule a little less mysterious. Keep practicing, and you'll be rotating polygons like a pro! If you have any question, feel free to ask.