3.14.2 Rotation
A rotation is a transformation that turns a figure around a fixed point by a certain angle. The figure is rotated clockwise or counterclockwise by the specified angle.
In other words, a rotation is a rigid motion that preserves the shape and size of the figure but changes its orientation.
To perform a rotation, you need to specify the angle of rotation and the center of rotation.
The angle of rotation can be measured in degrees or radians, and the center of rotation can be any point in the plane.
The direction of rotation can be clockwise or counterclockwise.
A rotation can be described using coordinates or vectors.
If the original figure has coordinates \( (x,y) \), then its rotated image will have coordinates \( (x’, y’) \).
For example, if you want to rotate a square \( 90 \) degrees counterclockwise around the origin, you would use the following formulas:
\( x’ = -y \)
\( y’ = x \)
This will rotate the entire square \( 90 \) degrees counterclockwise around the origin, without changing its shape or size.
Worked example:
On the grid, draw the image of
triangle \( A \) after a rotation of \( 90° \) anticlockwise about \( (0,0) \)
Mark the point with coordinate \( (0,0) \)
Put tracing paper over both the marked point and triangle \( A \).
Trace triangle \( A \) and then rotate tracing paper \( 90° \) anticlockwise, along the point \( (0,0) \) by holding it with the tip of your pen.
triangle \( A \) after a translation by the vector \( \frac{3}{-5} \)
A translation is a transformation that moves a figure in a straight line without changing its shape or orientation.
The top number in the vector is \( 3 \), so move the shape \( A \) \( 3 \) units to its right.
The bottom number is \( -5 \), so move the shape \( A \) \( 5 \) units downward.
