Transformation Geometry (Rotation)

In the IGCSE syllabus, we have rotation of objects through multiples of 90 degrees, that means we may have questions involving rotations of objects or geometrical shapes through an angle of 90 degrees, 180 degrees or 270 degrees. An object when rotated through an angle of 360 degrees returns to its original position and orientation.

If we consider a rotation of an object through an angle of 90 degrees, the direction in which the object would be aligned after rotation, would depend upon whether we are rotating it in the clockwise or in the anticlockwise direction. Therefore, whenever we specify a 90 degree rotation, we also specify the direction of rotation.

But if we rotate a shape or an object through 180 degrees, it gets aligned in the same direction irrespective of the direction of rotation, that is, it doesn't matter whether we move the object in the clockwise direction or in the anticlockwise direction. A 180 degree rotation can be thought of as rotating an object through an angle of 90 degrees twice in the same direction. A 180 degrees rotation can be interpreted in another way. We will find that at the end of this discussion.

A rotation of 270 degrees depends on the direction of rotation. But then, we realize that rotating an object through an angle of 270 degrees in the clockwise direction is equivalent to rotating that object through an angle of 90 degrees in the anticlockwise direction. Similarly, instead of rotating a figure through an angle of 270 degrees in the anticlockwise direction, we may rotate the figure through an angle of 90 degrees in the clockwise direction.

Therefore, practically, it comes to knowing how to rotate the shapes through an angle of 90 degrees both clockwise and anticlockwise.

In IGCSE, you mostly get two types of questions --

1. A shape, the angle of rotation along with the direction (if needed) and the center of rotation is given and the rotated shape has to be drawn.

2. Both the original and the rotated shapes are given and the angle of rotation along with the direction (if required) and the center of rotation has to be found out.

Let us take up both the types one by one.

1. Given: a) Original triangle ABC

b) 90 degrees rotation in the clockwise direction

c) Center of rotation (2, -2)

To Be Found: Rotated triangle A'B'C'

i) Given situation: The triangle ABC has been drawn and the center of rotation has been marked and named as point R.

ii) A rectangular handle has been drawn connecting the center of rotation R with the point A. Two options have been shown. The first one with the solid line and the second one with the dotted line. We may draw any of them.

iii) This rectangular handle has then been rotated through an angle of 90 degrees in the clockwise direction, making the point R the pivot of the rotation. The final end point of the handle has been marked A'.

iv) Similarly, the points R and B have been connected to form a rectangular handle which has then been rotated through an angle of 90 degrees in the clockwise direction, making the point R the pivot of rotation, to get the point B'. We may draw some other rectangular handle as well starting from the center of rotation R.

v) The same procedure has been repeated for getting the point C'.

vi) The points A', B' and C' have been finally joined to form the rotated triangle A'B'C'.

vii) For a 180 degrees rotation, the rectangular handles should have to be rotated through 180 degrees to get the rotated points.

2. Given: Both the original and the rotated triangles are given.

To Be Found: a) Angle and the direction of rotation

b) Center of rotation

i) The original triangle ABC and the rotated triangle A'B'C' have been given.

ii) By looking at the two triangles we understand that the triangle ABC has been rotated in the anticlockwise direction through an angle of 90 degrees in order to bring it to the position of the triangle A'B'C'.. It is not 180 degrees rotation because in case of 180 degrees rotation, the shape gets flipped.

iii) The points A and A' have been connected.

iv) The perpendicular bisector of the line segment AA' has been constructed using a ruler and a compass.

v) Similarly, the points B and B' have been connected and the perpendicular bisector of the line segment BB' has been drawn.

vi) The intersecting point of the two perpendicular bisectors has been marked as O. This is the center of rotation.

vii) We may draw the perpendicular bisector of the line segment CC' instead of the perpendicular bisector of the line segment AA' or BB'. The intersecting point of the perpendicular bisectors gives us the center of rotation.

viii) The coordinates of the center of rotation could be known if it would have been a grid paper or a graph paper. I apologize for this shortcoming.

This was all about the two types of questions related to rotation which are very common in the IGCSE board Math papers.

The third type involving 180 degrees rotation is very rarely asked but it is always better to be prepared for every type of question.

3. Given: Both the original and the rotated triangles are given where the transformation is 180 degrees rotation.

To Be Found: a) Center of rotation

Both triangles ABC and A'B'C' are given. We find that the transformation is 180 degrees rotation. But it can also be named as an enlargement transformation with scale factor -1. Therefore, these two transformations are equivalent and hence the center of rotation (or enlargement) can be found by getting the coordinates of the intersecting point of the line segments formed by joining the corresponding points of the object and the image. In our example, it is an 180 degrees rotation about the origin (0 , 0).

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