Rotate Rectangle: Find Original Vertex Q Coordinates

Hey there, math enthusiasts! Today, we're diving into a fascinating problem involving geometric transformations, specifically rotations. We'll be working with a rectangle that has been rotated, and our mission is to find the original location of one of its vertices. Get ready to put on your detective hats and unravel this mathematical puzzle!

The Challenge: Decoding the Rotation

Our problem presents us with a rectangle that has undergone a transformation known as $R_{0,90^{\circ}}$. This notation tells us that the rectangle has been rotated 90 degrees counterclockwise around the origin (0,0). We're given the coordinates of the transformed rectangle's vertices: $R'(-4,4)$, $S'(-4,1)$, $P'(-3,1)$, and $Q'(-3,4)$. The crucial question we need to answer is: what were the original coordinates of vertex Q before the rotation?

Before we jump into the solution, let's take a moment to understand what a 90-degree counterclockwise rotation does to a point's coordinates. This understanding is the key to solving this problem. When a point (x, y) is rotated 90 degrees counterclockwise around the origin, its image becomes (-y, x). In other words, the x and y coordinates switch places, and the new x-coordinate is the negative of the original y-coordinate. This transformation rule is the cornerstone of our solution strategy. So, keep this rule in mind as we continue our exploration.

Cracking the Code: Reversing the Rotation

Now that we understand the effect of a 90-degree counterclockwise rotation, we can reverse the process to find the original coordinates of point Q. We know that the image of Q after the rotation is $Q'(-3,4)$. To find the original coordinates of Q, we need to apply the inverse transformation. Since the rotation was 90 degrees counterclockwise, the inverse transformation would be a 90-degree clockwise rotation (or a 270-degree counterclockwise rotation).

However, there's a simpler way to think about this. We can reverse the coordinate transformation rule we discussed earlier. If a point (x, y) becomes (-y, x) after a 90-degree counterclockwise rotation, then to go back to the original point, we need to reverse this process. So, if we have the point (-y, x), we can find the original point by swapping the coordinates and negating the new y-coordinate. This gives us the original point (x, y). Let's apply this to our problem.

We have $Q'(-3, 4)$. To find the original coordinates of Q, we swap the coordinates to get (4, -3), and then negate the new y-coordinate to get (4, 3). Therefore, the original coordinates of Q are (4, 3). But wait! This answer isn't among the options provided. Let's re-examine our steps to ensure we haven't made any errors. It's crucial in math problems to double-check your work, guys, because a small mistake can lead to a completely different answer.

Let’s think step by step. We have $Q'(-3,4)$. Reversing the 90-degree counterclockwise rotation means applying a 90-degree clockwise rotation. The rule for a 90-degree clockwise rotation is (x, y) becomes (y, -x). Applying this rule to $Q'(-3, 4)$, we get (4, -(-3)), which simplifies to (4, 3). So, it seems our calculation is correct. However, the answer (4, 3) is not among the options. This indicates there might be an error in the provided options or in our interpretation of the question.

Since we've thoroughly checked our calculations and the logic is sound, it's possible that there's a mistake in the given answer choices. In such situations, it's important to highlight the discrepancy and, if possible, provide the correct answer based on our analysis. In this case, our analysis points to (4, 3) as the correct original coordinates of point Q.

Exploring the Geometry: Visualizing the Transformation

To solidify our understanding, let's visualize the transformation. Imagine the rectangle in the coordinate plane. The rotation $R_{0,90^{\circ}}$ will rotate the rectangle 90 degrees counterclockwise around the origin. If you were to plot the points $R'(-4,4)$, $S'(-4,1)$, $P'(-3,1)$, and $Q'(-3,4)$, you would see the rotated rectangle. Now, visualizing the reverse rotation, we can imagine rotating the rectangle 90 degrees clockwise back to its original position. This visualization can help confirm our calculated coordinates for the original point Q.

Moreover, we can use the other given points to verify our understanding of the transformation. For instance, if we apply the reverse rotation (90 degrees clockwise) to $R'(-4,4)$, we get (4, -(-4)) which simplifies to (4, 4). Similarly, applying the reverse rotation to $S'(-4,1)$ gives us (1, -(-4)) which simplifies to (1, 4), and applying it to $P'(-3,1)$ gives us (1, -(-3)) which simplifies to (1, 3). These calculations help us reconstruct the original rectangle and further validate our approach.

Remember, geometry is not just about formulas and calculations; it's also about visualizing shapes and transformations in space. Developing this visual intuition can greatly enhance your problem-solving skills in geometry.

Key Takeaways: Mastering Rotations

This problem highlights several important concepts related to geometric transformations:

1. Understanding Rotation Rules: Knowing how rotations affect coordinates is crucial. A 90-degree counterclockwise rotation around the origin transforms (x, y) into (-y, x). A 90-degree clockwise rotation transforms (x, y) into (y, -x).
2. Inverse Transformations: To undo a transformation, we apply its inverse. The inverse of a 90-degree counterclockwise rotation is a 90-degree clockwise rotation (or a 270-degree counterclockwise rotation).
3. Reversing the Coordinate Transformation Rule: Instead of memorizing separate rules for clockwise and counterclockwise rotations, we can reverse the basic transformation rule to find the original coordinates.
4. Visualizing Transformations: Visualizing the geometric transformations can aid in understanding and problem-solving.
5. Double-Checking and Critical Thinking: Always double-check your calculations and be prepared to question the given information if your results don't match the options.

By mastering these concepts, you'll be well-equipped to tackle a wide range of geometry problems involving rotations and other transformations. Keep practicing, and you'll become a transformation whiz in no time!

Final Thoughts: The Beauty of Geometric Transformations

Geometric transformations are a fundamental concept in mathematics, with applications in various fields, including computer graphics, physics, and engineering. Understanding transformations allows us to manipulate shapes and objects in space, creating fascinating patterns and designs. This problem, while seemingly simple, touches upon the core principles of rotations and inverse transformations. By successfully navigating this problem, we've not only honed our mathematical skills but also deepened our appreciation for the beauty and elegance of geometry.

So, keep exploring, keep questioning, and keep transforming your understanding of mathematics! You've got this!