Rotation 90 Degrees Counterclockwise About The Origin Worksheet
Table of Contents :
Understanding the concept of rotation in geometry is essential for mastering various mathematical problems, particularly in coordinate geometry. Today, we'll dive into the process of rotating points in the plane, specifically focusing on a 90-degree counterclockwise rotation about the origin. This transformation not only plays a crucial role in mathematics but is also applicable in fields such as computer graphics, physics, and engineering.
What is Rotation?
In geometry, rotation refers to turning a figure about a fixed point called the center of rotation. The angle of rotation determines how far to turn the figure. In our case, we will explore 90 degrees counterclockwise rotation around the origin (0,0).
Understanding the Coordinates
When you rotate a point in a Cartesian coordinate system, the coordinates of the point will change based on the angle of rotation. For a point ( P(x, y) ), the new coordinates ( P'(x', y') ) after a 90-degree counterclockwise rotation can be determined using the following transformation:
- Original Coordinates: ( P(x, y) )
- New Coordinates after 90-degree Counterclockwise Rotation:
- ( x' = -y )
- ( y' = x )
Visualization of Rotation
To better understand this transformation, let’s take a look at a few examples.
-
Example Point (2, 3):
- Before Rotation: ( P(2, 3) )
- After Rotation:
- ( x' = -3 )
- ( y' = 2 )
- New Coordinates: ( P'(-3, 2) )
-
Example Point (-1, 4):
- Before Rotation: ( P(-1, 4) )
- After Rotation:
- ( x' = -4 )
- ( y' = -1 )
- New Coordinates: ( P'(-4, -1) )
The Transformation Table
To assist in understanding the rotations better, we can create a transformation table that shows several points before and after a 90-degree counterclockwise rotation.
<table> <tr> <th>Original Point (x, y)</th> <th>New Point after Rotation (x', y')</th> </tr> <tr> <td>(2, 3)</td> <td>(-3, 2)</td> </tr> <tr> <td>(-1, 4)</td> <td>(-4, -1)</td> </tr> <tr> <td>(0, 5)</td> <td>(-5, 0)</td> </tr> <tr> <td>(3, -2)</td> <td>(2, 3)</td> </tr> <tr> <td>(-4, -3)</td> <td>(-3, 4)</td> </tr> </table>
Applying the Rotation in Practice
When applying this rotation, it’s essential to remember a few key points:
- Direction of Rotation: A counterclockwise rotation moves points in a circular direction around the origin.
- Negative and Positive Signs: The transformation formula involves changing signs; make sure to keep track of this to avoid mistakes.
Rotating Shapes
Rotating points individually is relatively straightforward, but what if you have a whole shape, like a triangle or a square? The process remains the same! Simply apply the rotation rules to each vertex of the shape.
Example: Triangle Rotation
Consider a triangle with vertices at (1, 1), (2, 3), and (3, 1). Let’s find the new coordinates after a 90-degree counterclockwise rotation.
-
Vertex A (1, 1):
- ( x' = -1 )
- ( y' = 1 )
- New Vertex A: (-1, 1)
-
Vertex B (2, 3):
- ( x' = -3 )
- ( y' = 2 )
- New Vertex B: (-3, 2)
-
Vertex C (3, 1):
- ( x' = -1 )
- ( y' = 3 )
- New Vertex C: (-1, 3)
Important Notes on Transformation
"When performing a rotation, ensure you visualize the process on graph paper or a coordinate plane to enhance understanding and accuracy."
Challenges and Exercises
To master the concept of rotation, try practicing with a variety of points and shapes. Here are a few exercises to consider:
- Rotate the point (4, 2) counterclockwise by 90 degrees.
- Take the square with corners at (1, 1), (1, -1), (-1, -1), (-1, 1) and find the new corners after rotation.
- What will be the new coordinates of the point (-2, -3) after a 90-degree counterclockwise rotation?
Conclusion
Rotating points and shapes is a fundamental skill in geometry that enhances your understanding of spatial relationships. By mastering the rules of 90-degree counterclockwise rotation about the origin, you pave the way for more complex geometrical concepts. Remember to practice regularly, and soon you will be rotating figures with ease! ✨