Congruency, Enlargement and Combined Transformations
How to Construct the image under a rotation
Use this to draw the image of a shape after a rotation about a given centre.
Before you start
- Plotting points on a coordinate grid
- Using compasses and a protractor
- The meaning of clockwise and anticlockwise
When to use it
Use this to draw the image of a shape after a rotation about a given centre.
The steps
- Mark the centre of rotation and note the angle and direction.
- Join the centre to each key vertex of the shape.
- Turn each of those lines through the given angle, in the given direction.
- Mark the new position of each vertex at the same distance from the centre.
- Join the new vertices to draw the image.
Worked example
Triangle T has vertices A(2, 1), B(5, 1) and C(2, 3). Construct the image of T under a rotation of 90° anticlockwise about the origin O(0, 0), and state the image vertices.
- Centre is O(0, 0); the angle is 90° and the direction is anticlockwise.
- Join O to each vertex A, B and C.
- Turn each line 90° anticlockwise; this sends a point (x, y) to (−y, x).
- Mark the images at the same distance from O: A(2, 1) → A′(−1, 2), B(5, 1) → B′(−1, 5), C(2, 3) → C′(−3, 2).
- Join A′, B′ and C′ to draw the image triangle.
A second example, with a twist
The centre is not the origin and the turn is clockwise, so distances are measured from a moved centre. Rotate the point D(4, 3) by 90° clockwise about the centre M(1, 1).
- Centre is M(1, 1); the angle is 90° and the direction is clockwise.
- Join M to D. The position of D relative to M is (4 − 1, 3 − 1) = (3, 2).
- A 90° clockwise turn sends a relative position (x, y) to (y, −x), so (3, 2) → (2, −3).
- Add the centre back to keep the same distance from M: D′ = (1 + 2, 1 + (−3)) = (3, −2).
- Mark D′ as the rotated point and join it to any figure it belongs to.
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
Which direction is positive, clockwise or anticlockwise?
In SPM a positive angle of rotation is anticlockwise and a negative angle is clockwise, matching the way angles grow on a graph. Always read the question for the word clockwise or a minus sign, because turning the wrong way puts the image in the wrong place.
How do I keep each vertex the same distance from the centre?
Use compasses. Put the point on the centre and the pencil on a vertex, then swing an arc; the image vertex lies on that arc at the required angle.
This keeps the distance unchanged, which is exactly what makes a rotation an isometry that preserves size and shape.
Can I rotate just the vertices instead of the whole shape?
Yes. Rotate each key vertex, then join the images in the same order.
Because a rotation keeps straight lines straight and lengths equal, joining the rotated vertices reproduces the whole shape correctly, so you never need to rotate every point along the edges.