Rotating Triangle LMN 90 Degrees Clockwise A Step-by-Step Solution

JU07/21/2025 08, 2025 by THE IDEN 67 views

Rotating Triangle LMN 90 Degrees Clockwise About the Origin

This article will explore the concept of rotating a triangle in the coordinate plane, specifically focusing on rotating triangle LMN 90 degrees clockwise about the origin. We will determine the coordinates of the vertices of the image L'M'N' after this transformation. This is a fundamental concept in geometry and is crucial for understanding transformations and their effects on geometric figures.

Understanding Rotations in the Coordinate Plane

Rotations in the coordinate plane involve turning a figure about a fixed point, which in this case is the origin (0,0). A 90-degree clockwise rotation is a specific type of transformation that alters the position of points while maintaining the shape and size of the figure. Understanding how coordinates change during rotations is essential for solving geometry problems and visualizing transformations.

When a point (x, y) is rotated 90 degrees clockwise about the origin, its new coordinates become (y, -x). This transformation rule is derived from the geometric properties of rotations and the way the coordinate axes are oriented. To fully grasp this rule, it's helpful to visualize the point's movement as it rotates around the origin. Imagine the point tracing a circular path, and consider how its x and y coordinates change as it moves through each quadrant.

This transformation rule is not just a formula to memorize; it represents a fundamental geometric principle. The rotation essentially swaps the x and y coordinates and negates the new y-coordinate. This negation reflects the change in direction as the point moves from one quadrant to another during the clockwise rotation. For instance, a point in the first quadrant (where both x and y are positive) will move to the fourth quadrant (where x is positive and y is negative) after a 90-degree clockwise rotation. This geometric interpretation provides a deeper understanding of the transformation and makes it easier to apply in various contexts.

Problem Statement: Triangle LMN and its Vertices

We are given triangle LMN with vertices at L(-1, 5), M(-1, 0), and N(-2, 5). Our goal is to determine the coordinates of the vertices of the image L'M'N' after rotating triangle LMN 90 degrees clockwise about the origin. This involves applying the rotation rule to each vertex of the triangle and finding its new position in the coordinate plane.

The given coordinates define the triangle's shape and position. Point L is located at (-1, 5), meaning it is 1 unit to the left of the origin and 5 units above it. Point M is at (-1, 0), situated on the negative x-axis, 1 unit to the left of the origin. Point N is at (-2, 5), 2 units to the left and 5 units above the origin. These coordinates provide the foundation for understanding how the triangle will transform under rotation.

To solve this problem, we will apply the 90-degree clockwise rotation rule to each of these vertices individually. This means we will take the coordinates of each point, apply the transformation (x, y) -> (y, -x), and then plot the new points to visualize the transformed triangle. By understanding the initial positions of the vertices and applying the rotation rule accurately, we can determine the precise location of the image L'M'N'. This process demonstrates the practical application of geometric transformations and their effects on coordinate points.

Applying the 90-Degree Clockwise Rotation Rule

To find the vertices of the image L'M'N', we will apply the 90-degree clockwise rotation rule (x, y) → (y, -x) to each vertex of the original triangle LMN.

Rotating Vertex L (-1, 5)

For vertex L (-1, 5), we apply the rule as follows:

x = -1, y = 5
L' (y, -x) = L' (5, -(-1)) = L' (5, 1)

Therefore, the image of vertex L after the rotation is L' (5, 1). This means that after the rotation, the new point L' is located 5 units to the right of the origin and 1 unit above it. This transformation reflects the 90-degree clockwise rotation, where the original x and y coordinates are swapped, and the new y-coordinate is the negation of the original x-coordinate.

Rotating Vertex M (-1, 0)

For vertex M (-1, 0), the rotation rule is applied similarly:

x = -1, y = 0
M' (y, -x) = M' (0, -(-1)) = M' (0, 1)

Thus, the image of vertex M after the rotation is M' (0, 1). This point lies on the positive y-axis, 1 unit above the origin. The transformation illustrates how a point on the negative x-axis rotates to the positive y-axis under a 90-degree clockwise rotation. The x-coordinate becomes the new y-coordinate, and the original y-coordinate becomes the new x-coordinate, with the appropriate negation applied.

Rotating Vertex N (-2, 5)

Lastly, for vertex N (-2, 5):

x = -2, y = 5
N' (y, -x) = N' (5, -(-2)) = N' (5, 2)

Hence, the image of vertex N after the rotation is N' (5, 2). This point is located 5 units to the right of the origin and 2 units above it. The transformation rule consistently swaps the coordinates and negates the appropriate value, demonstrating the predictable nature of rotations in the coordinate plane. By applying this rule to each vertex, we can accurately determine the transformed coordinates and visualize the rotated figure.

Solution: Vertices of Image L'M'N'

Based on the calculations above, the vertices of the image L'M'N' after rotating triangle LMN 90 degrees clockwise about the origin are:

L' (5, 1)
M' (0, 1)
N' (5, 2)

These new coordinates represent the transformed positions of the original vertices after the rotation. L' (5, 1) is the image of L (-1, 5), M' (0, 1) is the image of M (-1, 0), and N' (5, 2) is the image of N (-2, 5). By plotting these points on a coordinate plane, you can visualize the rotated triangle and confirm that it is indeed a 90-degree clockwise rotation of the original triangle LMN.

This solution demonstrates the application of the 90-degree clockwise rotation rule in transforming geometric figures. Understanding and applying such transformations is crucial in various fields, including computer graphics, engineering, and mathematics. The ability to accurately determine the new coordinates after a transformation allows for precise manipulation and analysis of geometric shapes and their properties.

Conclusion

In conclusion, by applying the 90-degree clockwise rotation rule to each vertex of triangle LMN, we have successfully determined the vertices of the image L'M'N'. The vertices of the rotated triangle are L'(5, 1), M'(0, 1), and N'(5, 2). This exercise illustrates the fundamental principles of geometric transformations and their effects on coordinate points.

Understanding rotations and other transformations is crucial in various mathematical and practical contexts. Rotations preserve the shape and size of the figure while changing its orientation. The 90-degree clockwise rotation is a specific example of this, and the rule (x, y) → (y, -x) provides a systematic way to determine the new coordinates after the transformation. This concept is not only important in geometry but also has applications in fields such as computer graphics, where objects need to be rotated and manipulated in a virtual space.

The ability to perform and understand geometric transformations is a valuable skill in problem-solving and spatial reasoning. It allows for the accurate manipulation of figures and the prediction of their positions after transformations. This exercise with triangle LMN serves as a clear example of how the rotation rule is applied and how the new coordinates are determined. By mastering these concepts, one can confidently tackle more complex geometric problems and applications.