Triangle Rotation A Coordinate Geometry Transformation Problem

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In the realm of coordinate geometry, understanding transformations is crucial. Transformations include rotations, reflections, translations, and dilations, each altering the position or size of a geometric figure while preserving certain properties. Here, we tackle a specific problem involving the rotation of a triangle in the coordinate plane. Our focus will be on understanding the rules of rotation and applying them to find the new coordinates of a point after transformation. Let’s dive into the problem and explore the concepts involved in detail.

The Problem Statement

The problem presents a triangle with vertices at specific coordinates: R(1, 1), S(-2, -4), and T(-3, -3). This triangle undergoes a transformation, specifically a rotation, denoted as R_{0, 270^{\circ}}. This notation signifies a rotation about the origin (0, 0) by an angle of 270 degrees in a counterclockwise direction. The task is to determine the new coordinates of the vertex S after this transformation, which we will denote as S'. The provided options are:

  • A. (-4, 2)
  • B. (-2, 4)
  • C. (2, 4)
  • D. (4, -2)

To solve this problem, we need to understand the rules governing rotations in the coordinate plane. A rotation of 270 degrees counterclockwise about the origin has a specific effect on the coordinates of a point. Let’s delve into the mechanics of rotations and how they affect coordinates.

Understanding Rotations in Coordinate Geometry

In coordinate geometry, a rotation is a transformation that turns a figure about a fixed point, known as the center of rotation. The rotation is defined by the angle of rotation and the direction (clockwise or counterclockwise). The most common center of rotation is the origin (0, 0), and we often deal with rotations of specific angles such as 90 degrees, 180 degrees, and 270 degrees. Each of these rotations has a specific rule that dictates how the coordinates of a point change after the transformation.

The general rule for a counterclockwise rotation of θ{\theta} degrees about the origin can be expressed using transformation matrices, but for specific angles like 90, 180, and 270 degrees, there are simpler rules we can apply directly. These rules are derived from the transformation matrices but are easier to remember and apply in problem-solving scenarios. For a 270-degree counterclockwise rotation, the rule is particularly straightforward:

Rule for 270-degree Counterclockwise Rotation

A 270-degree counterclockwise rotation about the origin transforms a point (x, y) to a new point (y, -x). This means that the original y-coordinate becomes the new x-coordinate, and the negation of the original x-coordinate becomes the new y-coordinate. This rule is crucial for solving the problem at hand. To illustrate, let’s consider a general point (x, y) and see how it transforms under a 270-degree counterclockwise rotation.

  • Original point: (x, y)
  • After 270-degree counterclockwise rotation: (y, -x)

This transformation essentially swaps the x and y coordinates and changes the sign of the new y-coordinate. This understanding is fundamental to solving problems involving rotations in coordinate geometry.

Visualizing the 270-degree Rotation

To further clarify, let's visualize what happens when a point is rotated 270 degrees counterclockwise about the origin. Imagine a point in the first quadrant. After a 90-degree counterclockwise rotation, it moves to the second quadrant. After 180 degrees, it lands in the third quadrant. Finally, after 270 degrees, it ends up in the fourth quadrant. The coordinates change accordingly, following the rule we discussed.

For example, consider the point (1, 2). After a 270-degree counterclockwise rotation:

  • The x-coordinate (1) becomes the new y-coordinate with a negative sign: -1
  • The y-coordinate (2) becomes the new x-coordinate: 2
  • The new point is (2, -1)

This visualization helps in grasping the transformation and confirming the rule we apply.

Applying the Rotation Rule to Find S'

Now that we have a firm understanding of the 270-degree counterclockwise rotation rule, we can apply it to the given problem. The vertex S has coordinates (-2, -4). We need to find the coordinates of S' after the rotation R_{0, 270^{\circ}}. Applying the rule (x, y) \rightarrow (y, -x), we have:

  • Original point S: (-2, -4)
  • After 270-degree counterclockwise rotation S': (-4, -(-2))
  • Simplifying, S': (-4, 2)

So, the new coordinates of S' are (-4, 2). This matches option A in the provided choices.

Step-by-Step Calculation

To ensure clarity, let’s break down the calculation step by step:

  1. Identify the coordinates of the original point S: (-2, -4)
  2. Apply the 270-degree counterclockwise rotation rule: (x, y) \rightarrow (y, -x)
  3. Substitute the coordinates of S into the rule:
    • New x-coordinate = y-coordinate of S = -4
    • New y-coordinate = - (x-coordinate of S) = - (-2) = 2
  4. The new coordinates of S' are (-4, 2)

This step-by-step approach ensures that we correctly apply the rotation rule and arrive at the correct answer. It also highlights the importance of paying close attention to the signs of the coordinates.

Verification and Alternative Approaches

To verify our solution, we can consider alternative approaches or use graphical methods. One way to visualize the rotation is to plot the original point S and then imagine rotating it 270 degrees counterclockwise about the origin. This visual approach can help confirm that the new coordinates should indeed be in the second quadrant, which aligns with our result of (-4, 2).

Graphical Verification

  1. Plot the original point S (-2, -4) on the coordinate plane. This point is in the third quadrant.
  2. Imagine rotating the point 90 degrees counterclockwise. It would move to the second quadrant.
  3. Rotate another 90 degrees (180 degrees total). It would move to the first quadrant.
  4. Rotate the final 90 degrees (270 degrees total). It would move to the second quadrant.
  5. The new point S' (-4, 2) is indeed in the second quadrant, which supports our calculated result.

Understanding the 90-degree Rotation Rule

Another way to approach this problem is to break down the 270-degree rotation into three 90-degree rotations. A 90-degree counterclockwise rotation about the origin transforms a point (x, y) to (-y, x). Applying this rule three times would achieve the same result as a single 270-degree rotation.

  1. First 90-degree rotation: (-2, -4) \rightarrow (4, -2)
  2. Second 90-degree rotation: (4, -2) \rightarrow (2, 4)
  3. Third 90-degree rotation: (2, 4) \rightarrow (-4, 2)

This approach also leads to the same coordinates for S', which further validates our solution.

Conclusion

In conclusion, the coordinates of S' after the 270-degree counterclockwise rotation are (-4, 2), which corresponds to option A. This problem highlights the importance of understanding the rules of rotations in coordinate geometry and applying them methodically. By breaking down the problem into smaller steps and using visualization techniques, we can confidently arrive at the correct solution. Transformations are a fundamental concept in geometry, and mastering them is essential for solving a wide range of problems.

By understanding the specific rules for rotations, such as the 270-degree counterclockwise rotation rule, we can efficiently solve problems involving transformations in the coordinate plane. The ability to visualize these transformations and apply the appropriate rules is a valuable skill in mathematics and related fields. Practice and a solid grasp of the underlying principles are key to success in this area.

This comprehensive exploration of the problem, the rotation rule, and the step-by-step solution provides a thorough understanding of the concepts involved. With this knowledge, similar problems can be approached with confidence and accuracy.