If Triangle ABC With Vertices A(-3,0), B(-2,3), And C(-1,1) Is Rotated 180 Degrees Clockwise About The Origin, Then Reflected Across The Line Y = -x, What Are The Coordinates Of The New Vertices?

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In the realm of geometry, understanding transformations is crucial. Geometric transformations involve altering the position, size, or orientation of a shape. These transformations include rotations, reflections, translations, and dilations. This article delves into a specific problem involving a triangle undergoing two transformations: a rotation and a reflection. We will explore how these transformations affect the coordinates of the vertices of a triangle, providing a step-by-step explanation to enhance your understanding of geometric manipulations.

Consider a triangle, denoted as △ABC\triangle ABC, with the following vertices:

  • A(-3, 0)
  • B(-2, 3)
  • C(-1, 1)

This triangle undergoes two successive transformations:

  1. A rotation of 180∘180^{\circ} clockwise about the origin.
  2. A reflection across the line y=−xy = -x.

The objective is to determine the final coordinates of the vertices of the transformed triangle. This problem combines two fundamental geometric transformations, requiring a clear understanding of how each affects the coordinates of points in the plane. Through a detailed exploration of each transformation, we will derive the final coordinates, reinforcing key principles of geometric transformations.

Step 1: Rotation of 180 Degrees Clockwise About the Origin

The first transformation involves rotating △ABC\triangle ABC by 180∘180^{\circ} clockwise about the origin. Rotation is a transformation that turns a figure about a fixed point, known as the center of rotation. In this case, the center of rotation is the origin (0, 0), and the angle of rotation is 180∘180^{\circ} clockwise. When a point (x, y) is rotated 180∘180^{\circ} about the origin, its coordinates change to (-x, -y). This transformation rule is crucial for understanding how the vertices of the triangle will be affected. To apply this transformation, we will apply this rule to each vertex of △ABC\triangle ABC.

  • Vertex A(-3, 0) rotated 180∘180^{\circ} becomes A'(-(-3), -0) = A'(3, 0). The x-coordinate changes from -3 to 3, while the y-coordinate remains 0 as it is rotated about the origin.
  • Vertex B(-2, 3) rotated 180∘180^{\circ} becomes B'(-(-2), -3) = B'(2, -3). The x-coordinate changes from -2 to 2, and the y-coordinate changes from 3 to -3. This rotation places B' in the fourth quadrant.
  • Vertex C(-1, 1) rotated 180∘180^{\circ} becomes C'(-(-1), -1) = C'(1, -1). The x-coordinate changes from -1 to 1, and the y-coordinate changes from 1 to -1. C' is now located in the fourth quadrant as well.

After the rotation, the new vertices of the triangle are A'(3, 0), B'(2, -3), and C'(1, -1). This rotation essentially flips the triangle across both the x and y axes. Understanding this rotation is a foundational step, as it sets the stage for the subsequent reflection transformation. The simplicity of the 180∘180^{\circ} rotation transformation rule allows for efficient calculation and clear visualization of the triangle's new position in the coordinate plane. This step illustrates the basic principles of rotational transformations in geometry, which are essential for further geometric problem-solving.

Step 2: Reflection Across the Line y = -x

The second transformation involves reflecting the rotated triangle across the line y=−xy = -x. Reflection is a transformation that creates a mirror image of a figure across a line, known as the line of reflection. In this case, the line of reflection is y=−xy = -x, a diagonal line passing through the origin with a slope of -1. When a point (x, y) is reflected across the line y=−xy = -x, its coordinates change to (-y, -x). This rule is crucial for understanding how the vertices of the rotated triangle will be affected by this reflection. To apply this transformation, we will apply this rule to each vertex of the rotated triangle, namely A'(3, 0), B'(2, -3), and C'(1, -1).

  • Vertex A'(3, 0) reflected across the line y=−xy = -x becomes A''(-0, -3) = A''(0, -3). The x-coordinate becomes the negative of the original y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate.
  • Vertex B'(2, -3) reflected across the line y=−xy = -x becomes B''(-(-3), -2) = B''(3, -2). Again, the x-coordinate becomes the negative of the original y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate.
  • Vertex C'(1, -1) reflected across the line y=−xy = -x becomes C''(-(-1), -1) = C''(1, -1). Here, the x-coordinate becomes the negative of the original y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate, resulting in the point remaining unchanged.

After the reflection, the final vertices of the transformed triangle are A''(0, -3), B''(3, -2), and C''(1, -1). This reflection across the line y=−xy = -x has swapped and negated the coordinates, effectively creating a mirror image of the rotated triangle. Understanding reflection transformations is vital in geometry, as it helps in visualizing spatial relationships and symmetries. This step completes the transformation process, providing the final coordinates of the triangle's vertices after both the rotation and the reflection.

After performing the rotation of 180∘180^{\circ} clockwise about the origin and the reflection across the line y=−xy = -x, the final coordinates of the vertices of the transformed triangle are:

  • A''(0, -3)
  • B''(3, -2)
  • C''(1, -1)

These coordinates represent the culmination of both transformations applied sequentially. Understanding the effect of each transformation individually and then combining them allows for a clear and systematic solution to geometric problems. This process underscores the importance of mastering individual transformation rules and their cumulative effects on geometric figures.

Visualizing the Transformations

To better understand the transformations, visualizing them on a coordinate plane can be highly beneficial. Initially, △ABC\triangle ABC with vertices A(-3, 0), B(-2, 3), and C(-1, 1) is plotted. The rotation of 180∘180^{\circ} about the origin moves the triangle to a new position with vertices A'(3, 0), B'(2, -3), and C'(1, -1). This rotation can be seen as flipping the triangle across both axes. Then, the reflection across the line y=−xy = -x creates a mirror image of the rotated triangle, resulting in the final vertices A''(0, -3), B''(3, -2), and C''(1, -1). Visualizing these transformations helps to solidify the understanding of how points move and how shapes change under different transformations.

Implications and Applications

The concepts of rotation and reflection are fundamental in various fields, including:

  • Computer Graphics: These transformations are used extensively in creating and manipulating images and objects in computer graphics.
  • Physics: Rotations and reflections play a crucial role in understanding symmetries and transformations in physical systems.
  • Engineering: Engineers use these principles in design and analysis, particularly in fields such as structural engineering and robotics.
  • Mathematics: Geometric transformations are a core topic in geometry, providing a foundation for more advanced concepts such as linear algebra and group theory.

Understanding these transformations not only helps in solving geometric problems but also provides a foundation for applications in various scientific and technological domains. The ability to visualize and mathematically describe these transformations is a valuable skill in many disciplines.

In summary, by applying a 180∘180^{\circ} clockwise rotation about the origin followed by a reflection across the line y=−xy = -x, we have successfully transformed △ABC\triangle ABC to a new position with vertices A''(0, -3), B''(3, -2), and C''(1, -1). This problem illustrates the importance of understanding geometric transformation rules and their sequential application. Mastering these transformations is essential for anyone studying geometry and related fields. The step-by-step approach outlined in this article provides a clear methodology for solving similar problems, emphasizing the significance of both rotational and reflection transformations in geometric manipulations.

By breaking down the problem into distinct steps and thoroughly explaining the transformation rules, this article aims to enhance your understanding of geometric transformations and their applications. The combination of rotation and reflection transformations is a common theme in geometry, and a solid grasp of these concepts will undoubtedly aid in tackling more complex geometric challenges. Through visualization, practical application, and theoretical understanding, the principles of geometric transformations can be effectively mastered and applied in various contexts.