Rotation
Geometry · Rotation
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 6 Rotation
Lesson objectives
By the end of this topic, you should be able to:
- State the properties of rotation as a transformation.
- Apply the properties of rotation in the Cartesian plane.
- Identify the centre and angle of rotation given an object and its image.
- Identify the point of rotational symmetry and state the order of rotational symmetry.
Rotation
A rotation turns a figure about a fixed point through a given angle.
a) What defines a rotation
Three things are needed: the centre, the angle, and the direction.
Anticlockwise is taken as positive; clockwise as negative.
Every point moves along a circular arc about the centre.
b) Properties
Each point stays the same distance from the centre as its image.
Rotation preserves length and angle, so the image is congruent to the object.
It preserves orientation, unlike reflection. Clockwise lettering stays clockwise.
The centre itself is invariant. It is the only point that does not move.
c) Rotation in the Cartesian plane
About the origin, these four results cover the standard cases.
Note that 90° clockwise and 270° anticlockwise give the same image. So do 180° in either direction.
Always state the direction unless the angle is 180°.
d) Finding the centre of rotation
Given an object and its image, join each point to its image.
Construct the perpendicular bisector of each joining line.
Where the bisectors cross is the centre of rotation.
Two pairs of points are enough. Use a third to check.
e) Finding the angle
Join the centre to a point and to its image.
Measure the angle between those two lines.
Note the direction of turn as well as the size.
f) Rotational symmetry
A figure has rotational symmetry if it looks unchanged after a rotation of less than a full turn.
The order of rotational symmetry counts the positions in which it looks the same during one full turn.
| Figure | Order |
|---|---|
| Equilateral triangle | 3 |
| Square | 4 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Parallelogram | 2 |
| Circle | infinite |
A regular polygon with n sides has order n, and the angle between positions is 360/n.
Every figure has order at least 1, since a full turn always restores it. Order 1 means no rotational symmetry in practice.
g) Where this is used
Gears and turbines. Designing logos and tiles. Analysing crystal structure. Any repeating circular pattern.
Words to know
- Rotation -- a transformation turning a figure about a fixed point through a given angle.
- Centre of rotation -- the fixed point about which the turning takes place.
- Rotational symmetry -- the property of looking unchanged after a partial turn.
- Order of rotational symmetry -- the number of matching positions in one full turn.
- Invariant point -- a point that does not move under a transformation.
:::checkpoint Check yourself
- Find the image of (2, 5) after a 90° anticlockwise rotation about the origin.
- Find the image of (−3, 4) after a 180° rotation about the origin.
- What is the order of rotational symmetry of a regular octagon?
- How do you find the centre of a rotation from an object and its image? :::
Bridge to practice
The exercises begin with the defining properties and the direction convention, move through coordinate rotations and locating the centre, and finish with rotational symmetry and order. For every rotation you describe, check that you have stated all three of centre, angle and direction.