Rotation

In the world of geometry, rotation is one of the key concepts of transformation. Transformations are changes you can make to shapes that move them into a new position. There are different types of transformations, including translation, reflection, dilation, and rotation, which is what we're focusing on here.

What is rotation?

A rotation in geometry is a transformation that rotates a figure around a fixed point. This fixed point is called the center of rotation. During a rotation, the entire figure moves in a circular path on a plane, and every point of the figure rotates through the same angle around the center.

This change is marked by three main components:

• Center of rotation: The point around which the shape rotates. This can be a point on, inside, or outside the shape.
• Rotation angle: This is the angle you rotate the shape around the center. It is usually measured in degrees (°). Common rotations include 90°, 180°, and 270°.
• Direction of rotation: It can be clockwise or anticlockwise.

Visualization of rotation

Let's look at a simple demonstration of rotation using the example of a simple triangle:

Triangle ABC: A (-1, -1) B (0, 1) C (2, 0)

If we rotate triangle ABC 90 degrees counterclockwise about the origin (0,0), then the new coordinates of the triangle will be:

New Triangle A'B'C': A' (1, -1) B' (-1, 0) C' (0, 2)

In this example, the blue triangle shows the original position, and the red dashed triangle shows the triangle after a 90° counterclockwise rotation around the origin.

Properties of rotation

There are several important properties of rotation that you should know:

• Preserves distance: A rotation is a symmetric transformation, which means it preserves the distance between any two points in the figure. Therefore, the shape of the figure does not change.
• Preserves orientation: This means that the order of points in the rotated shape is maintained, clockwise or counterclockwise, as in the original.
• Preserves angles: The angles between lines in a figure remain the same after rotation.
• Rotational symmetry: If a figure looks the same after partial rotation of less than a full circle (360°), then it has rotational symmetry.

Mathematical representation: rotation using matrices

Rotations can also be represented using matrices in the coordinate plane, which is particularly useful when dealing with transformations in algebraic form. The rotation matrix for an angle θ in the counterclockwise direction is given by:

| cos(θ) -sin(θ) |
| sin(θ) cos(θ) |

If you apply this matrix to a point or vector (x, y), you get the new rotation position (x', y'):

x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)

As an example, consider rotating a point (1, 1) 90° counterclockwise:

θ = 90°
cos(90°) = 0
sin(90°) = 1
x' = 1 * 0 - 1 * 1 = -1
y' = 1 * 1 + 1 * 0 = 1
New point after rotation: (-1, 1)

Challenges and exercises with rotation

When learning about rotations, it can be useful to practice with different shapes and centers of rotation. Here are some challenges you can try:

• Rotate a rectangle 180° around one of its vertices and watch the change.
• Take any polygon and rotate it 120° clockwise about a point away from the shape.
• Try to find the center of rotation for a given rotating shape and its pre-image.

Practice problem: Given the point (3, 4), rotate it 270° counterclockwise about the origin. Find the new coordinates of the point.

θ = 270°
cos(270°) = 0
sin(270°) = -1
x' = 3 * 0 - 4 * (-1) = 4
y' = 3 * (-1) + 4 * 0 = -3
New coordinates: (4, -3)

Explore rotational symmetry

Rotational symmetry is an interesting subtopic of rotation. A figure is said to have rotational symmetry if it can be rotated through an angle less than 360° and appears unchanged. Let's take the example of a regular hexagon:

If you rotate a regular hexagon 60° around its center, it will look exactly the same because of its symmetry. This is also true for rotations at 120°, 180°, 240°, and 300°.

The orange lines show possible axes of rotational symmetry for a regular hexagon.

Conclusion

Rotation is a fascinating transformation in geometry that rotates shapes around a fixed point, preserving their size, shape, and angular measure. It is an essential concept not only for understanding the fundamentals of geometry but also for exploring more complex topics such as symmetry and algebraic transformations.

By practicing and applying transformation rules for rotation, you can gain a deeper understanding of how geometric shapes and figures interact within a plane. As you progress, you may also encounter more complex applications of rotation in fields such as graphics, architecture, and even robotics, where it is important to understand how objects move and orient themselves.

Happy learning and exploring rotations!

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