Rotating Points: Find Q' After -270° Rotation

Have you ever wondered what happens when you spin a point around the origin on a coordinate plane? It's a fascinating concept in geometry called rotation! In this article, we'll explore how to determine the new location of a point after it's been rotated. Specifically, we'll tackle the problem of finding the coordinates of point Q' after rotating point Q(-5, -9) -270 degrees about the origin. This might sound a bit intimidating, but don't worry, we'll break it down step-by-step so you can easily understand the process and confidently solve similar problems. So, let's dive in and unravel the mystery of point rotations!

Understanding Rotations in the Coordinate Plane

Before we jump into the specific problem, let's take a moment to understand the basics of rotations in the coordinate plane. When we rotate a point, we're essentially spinning it around a fixed point, which in this case is the origin (0, 0). The amount of rotation is measured in degrees, and the direction can be either clockwise or counterclockwise. A positive angle indicates a counterclockwise rotation, while a negative angle indicates a clockwise rotation. This is a crucial distinction because the direction of rotation affects the final position of the point. Imagine spinning a wheel – rotating it 90 degrees clockwise is different from rotating it 90 degrees counterclockwise. The same principle applies to points in the coordinate plane. Now, consider the quadrants of the coordinate plane. The first quadrant has both x and y coordinates positive (+, +), the second quadrant has a negative x and a positive y (-, +), the third quadrant has both negative coordinates (-, -), and the fourth quadrant has a positive x and a negative y (+, -). As we rotate a point, it moves from one quadrant to another, and the signs of its coordinates change accordingly. Understanding how these signs change is key to accurately determining the new coordinates after a rotation. Furthermore, certain rotations have predictable effects on the coordinates. For instance, a 90-degree counterclockwise rotation swaps the x and y coordinates and negates the new x-coordinate. A 180-degree rotation negates both coordinates. And a 270-degree counterclockwise rotation swaps the coordinates and negates the new y-coordinate. We will use these rules as we solve our problem. Mastering these basic rotation rules will provide you with a solid foundation for tackling more complex problems involving geometric transformations. So, keep these concepts in mind as we move forward and apply them to find the location of Q' after its rotation.

The Rotation Rule for -270 Degrees

Now, let's focus on the specific rotation we're dealing with: -270 degrees. A -270-degree rotation means we're rotating the point 270 degrees clockwise around the origin. But here's a clever trick: rotating a point -270 degrees is the same as rotating it 90 degrees counterclockwise! This is because a full circle is 360 degrees, so -270 degrees is simply 90 degrees short of a full clockwise rotation. Thinking of it as a 90-degree counterclockwise rotation makes the problem easier to visualize and solve. We already know how to deal with 90 degrees counterclockwise rotations, right? The general rule for a 90-degree counterclockwise rotation is this: if you have a point (x, y), its image after the rotation will be (-y, x). This rule is derived from the geometric properties of rotation and the way the coordinate plane is structured. When a point is rotated 90 degrees counterclockwise, its x-coordinate becomes the new y-coordinate (with a sign change), and its y-coordinate becomes the new x-coordinate. This rule is your key to quickly and accurately finding the coordinates of a point after a -270-degree rotation. It saves you from having to visualize the rotation or use complex trigonometric calculations. To solidify your understanding, let's consider a simple example. If we have the point (2, 3) and rotate it 90 degrees counterclockwise, the new point will be (-3, 2). Notice how the x and y coordinates swapped places, and the original y-coordinate became negative. This same principle applies to any point you rotate 90 degrees counterclockwise (or -270 degrees). So, with this rule in hand, we're well-equipped to solve our main problem: finding the location of Q' after rotating Q(-5, -9) -270 degrees.

Applying the Rule to Point Q(-5, -9)

Okay, we've got the rule for a -270-degree (or 90-degree counterclockwise) rotation: (x, y) becomes (-y, x). Now it's time to put this rule into action and find the new coordinates of point Q after its rotation. Our point Q has the coordinates (-5, -9). This means x = -5 and y = -9. To find Q', we simply substitute these values into our rotation rule. So, Q' will have the coordinates (-(-9), -5). Notice how we've swapped the x and y values and negated the new x-coordinate, as the rule dictates. Now, let's simplify those coordinates. -(-9) is the same as +9, so the x-coordinate of Q' is 9. The y-coordinate remains -5. Therefore, the coordinates of Q' after the -270-degree rotation are (9, -5). Isn't it amazing how a simple rule can help us solve a seemingly complex problem? By understanding the rotation rule and applying it systematically, we've successfully found the new location of Q'. This demonstrates the power of mathematical rules and how they can make geometric transformations much easier to handle. Now that we have the coordinates of Q', let's take a look at the answer choices provided and see which one matches our result. This will confirm our solution and reinforce our understanding of the rotation process. So, let's move on and compare our answer with the given options.

Selecting the Correct Answer

We've determined that the coordinates of Q' after rotating Q(-5, -9) -270 degrees about the origin are (9, -5). Now, let's compare this result with the answer choices provided:

A. Q'(5, 9) B. Q'(-5, 9) C. Q'(9, -5) D. Q'(9, 5)

By carefully examining the options, we can see that option C, Q'(9, -5), perfectly matches our calculated coordinates. This confirms that we have correctly applied the rotation rule and found the accurate location of Q'. The other options have different combinations of signs and coordinate values, which means they are incorrect. Option A has both coordinates positive, while our y-coordinate is negative. Option B has a negative x-coordinate, which is also incorrect. And option D has a positive y-coordinate, which doesn't match our result either. Therefore, we can confidently select option C as the correct answer. This process of comparing our calculated result with the given options is a crucial step in problem-solving. It helps us verify our work and ensure that we haven't made any errors along the way. It also reinforces our understanding of the problem and the concepts involved. Now that we've successfully found the correct answer, let's take a moment to recap the steps we took and highlight the key concepts involved in solving this rotation problem. This will help solidify your understanding and prepare you to tackle similar problems in the future.

Conclusion

In this article, we've successfully found the new coordinates of point Q' after rotating point Q(-5, -9) -270 degrees about the origin. We started by understanding the basics of rotations in the coordinate plane, including the concept of positive and negative angles and how they relate to clockwise and counterclockwise rotations. Then, we focused on the specific rotation of -270 degrees, recognizing that it's equivalent to a 90-degree counterclockwise rotation. This allowed us to apply the rotation rule for 90 degrees counterclockwise, which states that a point (x, y) becomes (-y, x) after the rotation. By substituting the coordinates of Q into this rule, we found that Q' has the coordinates (9, -5). Finally, we compared our result with the given answer choices and confidently selected the correct answer, which was option C. This problem demonstrates the power of understanding and applying mathematical rules to solve geometric problems. By breaking down the problem into smaller steps and using a systematic approach, we were able to find the solution efficiently and accurately. Remember, practice is key to mastering these concepts. Try working through similar rotation problems with different points and rotation angles to solidify your understanding. And don't hesitate to review the concepts and rules we've discussed in this article whenever you need a refresher. Happy rotating! For further exploration of rotations and transformations in geometry, you might find helpful resources on websites like Khan Academy's geometry section.