Rotation Transformation



In this lesson, we will learn

  • what is rotation
  • how to draw the rotated image of an object given the center, the angle and the direction of rotation.
  • how to find the angle of rotation given the object, its image and the center of rotation.
  • how to rotate points and shapes on the coordinate plane about the origin.
Related Topics: More Geometry Lessons

What is Rotation?

A rotation is a transformation in which the object is rotated about a fixed point. The direction of rotation can be clockwise or anticlockwise.

The fixed point in which the rotation takes place is called the centre of rotation. The amount of rotation made is called the angle of rotation.

Example:

For any rotation, we need to specify the center, the angle and the direction of rotation.

Drawing The Rotated Image

Given the center of rotation and the angle of rotation we can determine the rotated image of an object.

Example:

Determine the image of the straight line XY under an anticlockwise rotation of 90˚ about O.


Solution:

Step 1: Join point X to O.

Step 2: Using a protractor, draw a line 90˚ anticlockwise from the line OX. Mark on the line the point X’ such that the line of OX = OX’

Step 3: Repeat steps 1 and 2 for point Y. Join the points and Y’ to form the line X’Y’.



The Angle Of Rotation

Given an object, its image and the centre of rotation, we can find the angle of rotation using the following steps.

Step 1 : Choose any point in the given figure and join the chosen point to the centre of rotation.

Step 2 : Find the image of the chosen point and join it to the centre of rotation.

Step 3 : Measure the angle between the two lines. The sign of the angle depends on the direction of rotation. Anti-clockwise rotation is positive and clockwise rotation is negative.

 

Example :

Figure A’B’C’ is the image of figure ABC. O is the centre of rotation. Find the angle of rotation.


Solution :

Step 1: Join A to O

Step 2: Join to O.

Step 3: Measure the angle AOA’

The angle of rotation is 62˚ anticlockwise or +62˚





This video covers the manual rotation of a polygon about a given point at a given angle. You will need a straightedge, a protractor, and a compass. We will perform rotations about a point inside the figure, one outside the figure and one on the figure.



How to rotate a figure around a fixed point using a compass and protractor.



Rotate points on the coordinate plane

We will now look at how points and shapes are rotated on the coordinate plane. It will be helpful to note the patterns of the coordinates when the points are rotated about the origin at different angles.

Geometry Rotation
A rotation is an isometric transformation: the original figure and the image are congruent. The orientation of the image also stays the same, unlike reflections. To perform a geometry rotation, we first need to know the point of rotation, the angle of rotation, and a direction (either clockwise or counterclockwise). A rotation is also the same as a composition of reflections over intersecting lines.

The following videos show clockwise and anticlockwise rotation of 0˚, 90˚, 180˚ and 270˚about the origin (0, 0). The pattern of the coordinates are also explored.









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