Rotating Triangle RST Finding Coordinates Of S After 210 Degree Rotation

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In the realm of geometry, transformations play a crucial role in understanding how shapes and figures behave in space. One such transformation is rotation, where a figure is turned about a fixed point. In this article, we'll explore the concept of rotating a triangle and, in particular, how to find the new coordinates of a vertex after a rotation. We'll delve into a specific example where a triangle with vertices R(1,1), S(-2,-4), and T(-3,-3) undergoes a rotation of 210 degrees about the origin. Our goal is to determine the coordinates of the transformed vertex S', which will provide valuable insights into the application of rotation transformations in geometric problems.

Understanding Rotations in Geometry

Rotations are fundamental transformations in geometry, involving turning a figure about a fixed point, known as the center of rotation. The amount of rotation is measured in degrees, and the direction of rotation is either clockwise or counterclockwise. A positive angle indicates a counterclockwise rotation, while a negative angle indicates a clockwise rotation. Understanding rotations is crucial in various fields, including computer graphics, robotics, and physics, where objects are manipulated and their positions and orientations need to be accurately determined.

In the Cartesian plane, rotations are often performed about the origin (0,0). The coordinates of a point (x,y) after a rotation of θ degrees about the origin can be found using the following rotation formulas:

x′=xcos⁡θ−ysin⁡θx' = x \cos θ - y \sin θ

y′=xsin⁡θ+ycos⁡θy' = x \sin θ + y \cos θ

These formulas are derived from trigonometric principles and provide a precise way to calculate the new coordinates of a point after rotation. To effectively use these formulas, one must have a solid grasp of trigonometric functions and their values for various angles. The application of these formulas is not just limited to simple rotations; they form the basis for understanding more complex transformations and spatial manipulations in geometry.

The Problem: Rotating Triangle RST

Consider a triangle RST with vertices R(1,1), S(-2,-4), and T(-3,-3). This triangle is subjected to a rotation of 210 degrees about the origin. Our task is to find the coordinates of the transformed vertex S', which is the image of vertex S after the rotation. This problem provides an excellent opportunity to apply the rotation formulas and understand how coordinates change under rotational transformations. The key to solving this problem lies in the correct application of the rotation formulas, ensuring the proper substitution of the given coordinates and the angle of rotation. This exercise not only reinforces the understanding of rotation transformations but also highlights the importance of precise calculations in geometric problems.

Applying the Rotation Formulas

To find the coordinates of S' after a 210-degree rotation, we apply the rotation formulas to the coordinates of S(-2,-4). The angle of rotation θ is 210 degrees. We need to calculate $ \cos 210° $ and $ \sin 210° $ to use in the formulas.

The angle 210 degrees lies in the third quadrant, where both cosine and sine are negative. We can express 210 degrees as 180 degrees + 30 degrees. Therefore:

cos⁡210°=cos⁡(180°+30°)=−cos⁡30°=−32 \cos 210° = \cos (180° + 30°) = -\cos 30° = -\frac{\sqrt{3}}{2}

sin⁡210°=sin⁡(180°+30°)=−sin⁡30°=−12 \sin 210° = \sin (180° + 30°) = -\sin 30° = -\frac{1}{2}

Now, we substitute the coordinates of S(-2,-4) and the values of $ \cos 210° $ and $ \sin 210° $ into the rotation formulas:

x′=(−2)⋅(−32)−(−4)⋅(−12)=3−2 x' = (-2) \cdot (-\frac{\sqrt{3}}{2}) - (-4) \cdot (-\frac{1}{2}) = \sqrt{3} - 2

y′=(−2)⋅(−12)+(−4)⋅(−32)=1+23 y' = (-2) \cdot (-\frac{1}{2}) + (-4) \cdot (-\frac{\sqrt{3}}{2}) = 1 + 2\sqrt{3}

Thus, the coordinates of S' are $(\sqrt{3} - 2, 1 + 2\sqrt{3})$. This calculation demonstrates the practical application of rotation formulas in determining the new position of a point after a geometric transformation.

Analyzing the Result

The coordinates of S' after the 210-degree rotation are $(\sqrt{3} - 2, 1 + 2\sqrt{3})$. This result provides a precise location of the transformed vertex S' in the Cartesian plane. It's important to note that the new coordinates are not simple integers, which is typical when dealing with rotations that are not multiples of 90 degrees. The presence of square roots in the coordinates indicates the influence of the trigonometric values of the rotation angle. Analyzing the result in the context of the original triangle and the rotation helps in visualizing the transformation and understanding how the triangle has been repositioned. This process of analysis is crucial in geometry for verifying the correctness of calculations and gaining a deeper understanding of the transformations involved.

Conclusion

In this article, we successfully determined the coordinates of S' after rotating triangle RST by 210 degrees about the origin. We utilized the rotation formulas, which are fundamental tools in geometry for understanding transformations. By applying these formulas and performing the necessary calculations, we found the coordinates of S' to be $(\sqrt{3} - 2, 1 + 2\sqrt{3})$. This exercise highlights the importance of understanding rotations and their mathematical representation in solving geometric problems. Furthermore, it reinforces the connection between geometry and trigonometry, demonstrating how trigonometric functions play a vital role in describing and analyzing spatial transformations. The ability to perform and analyze rotations is essential in various fields, making this a valuable concept to master.

Final Answer: The final answer is not among the options provided. The correct coordinates of S' after the rotation are $(\sqrt{3} - 2, 1 + 2\sqrt{3})$.