Turning about a point
A rotation spins a shape around a fixed point, the way a hand sweeps around the centre of a clock. Every point stays the same distance from that centre, it just swings around to a new position. The shape does not change size, so the image is congruent to the original, it simply faces a new way.
- Rotation
- A transformation that turns a shape through a given angle about a fixed point called the centre of rotation. Size and shape are unchanged.
The three things to state
To describe a rotation fully, you must give all three of these:
| What | Example |
|---|---|
| Angle of turn | \(90^\circ\), \(180^\circ\), \(270^\circ\) |
| Direction | clockwise or anticlockwise |
| Centre of rotation | a point, such as the origin \((0,0)\) |
A 180° turn needs no direction
For \(180^\circ\) the result is the same whether you turn clockwise or anticlockwise, so you can leave the direction out. For \(90^\circ\) and \(270^\circ\) the direction matters and must be stated.
Rules for common turns
About the origin, these coordinate rules save time. Turn the point \((x,y)\) as follows:
If you have no rule to hand, tracing paper is allowed: draw the shape, pin the centre with your pencil, and turn the paper by the angle.
A worked example
Rotate the point \(P(4,2)\) by \(90^\circ\) anticlockwise about the origin. Find its image.
Where this is assessed
Applying the coordinate rule is Criterion A (Knowing and understanding). A description that names all three parts, the angle, the direction and the centre, is Criterion C (Communicating), and leaving any one out loses a mark. Reading rotations in patterns, logos or engineering contexts reaches Criterion D (Applying maths in context).
Check yourself
1. Rotate the point (5,0) by 180° about the origin. Where does it land? +
The rule for \(180^\circ\) is \((x,y) \to (-x,-y)\). So \((5,0) \to \mathbf{(-5,0)}\), straight across to the other side of the origin.
2. Rotate the point (1,3) by 90° clockwise about the origin. Where does it go? +
The rule for \(90^\circ\) clockwise is \((x,y) \to (y,-x)\). So \((1,3) \to \mathbf{(3,-1)}\).
3. What three things must a full description of a rotation include? +
The angle of turn, the direction (clockwise or anticlockwise), and the centre of rotation. For a \(180^\circ\) turn the direction can be left out because both directions give the same image.
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