Geometric Transformations
This topic covers how to manipulate shapes on a coordinate grid using four key transformations: translation, reflection, rotation, and enlargement. Mastering these is crucial for questions involving geometric reasoning and coordinate geometry without a calculator.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Translations, reflections, and rotations are 'rigid' transformations; they produce a congruent image which has the same size and shape as the original object.
- Enlargement is a 'non-rigid' transformation; it produces a similar image which has the same shape and angles, but a different size, unless the scale factor is 1 or -1.
- A translation is fully described by a 2D vector, such as (x, y), indicating the horizontal and vertical shift.
- A rotation is defined by three components: a centre of rotation, an angle, and a direction (positive is anticlockwise by convention).
- An enlargement is defined by a centre of enlargement and a scale factor. A fractional scale factor (between -1 and 1) makes the shape smaller.
- A negative scale factor in an enlargement inverts the object through the centre of enlargement, placing the image on the opposite side.
Formulae
ImagePoint = Centre + k × (ObjectPoint - Centre) To calculate the coordinates of a point after an enlargement with scale factor k about a centre that is not the origin. All points are treated as position vectors.
Pnew(x, y) = (xold + a, yold + b) To find the new coordinates of a point after a translation by the vector (a, b).
Definitions
- Object
- The original shape before any transformation is applied.
- Image
- The resulting shape after a transformation has been applied to the object.
- Congruent
- Two shapes are congruent if they are identical in size and shape. One can be mapped onto the other using only translations, reflections, or rotations.
- Similar
- Two shapes are similar if they have the same shape but may be different sizes. All corresponding angles are equal and the ratio of corresponding side lengths is constant.
- Invariant Point
- A point that remains in the same position after a transformation. For example, the centre of rotation is an invariant point.
Worked example
Triangle ABC has vertices A(2, 4), B(2, 1), and C(4, 1). The triangle is rotated 90° clockwise about the origin, then translated by the vector (-3, 2). What are the coordinates of the final image of vertex A?
- 1
First, perform the rotation.
A 90° clockwise rotation about the origin transforms a point (x, y) to (y, -x).
- 2
Apply this rule to vertex A(2, 4).
The new coordinates, A', will be (4, -2).
- 3
Next, perform the translation on the rotated point A'.
The translation vector is (-3, 2).
- 4
Add the vector to the coordinates of A':
(4 + (-3), -2 + 2).
- 5
The final coordinates of the image of vertex A, let's call it A'', are (1, 0).
Answer: (1, 0)
Common mistakes
- ×Sign errors with negative scale factors: Forgetting that a negative scale factor both scales and inverts the object through the centre of enlargement. This often leads to incorrect signs in the final coordinates.
- ×Mistakes in rotation direction: Confusing clockwise and anticlockwise rotations. Remember that anticlockwise is the positive direction by convention.
- ×Incorrect centre for enlargement: When the centre of enlargement is not the origin, you must calculate vectors from this centre. Simply multiplying the object's coordinates by the scale factor is a common error.
- ×Mixing up coordinates in reflections: For reflections in diagonal lines like y=x or y=-x, it's easy to swap x and y coordinates incorrectly or forget a sign change.
No-calculator tips
- ✓Always sketch the transformation. A quick diagram on the coordinate axes can immediately show if your calculated image is in the wrong quadrant or is the wrong orientation, helping you spot sign errors.
- ✓For 90° rotations about a point C, treat the vector from C to your point P as a 'step'. For a 90° anticlockwise rotation, if the step is (Δx, Δy), the new step from C will be (-Δy, Δx). This avoids visual estimation.
- ✓Break down enlargements from a non-origin centre into vector steps: 1. Find the vector from the centre to your point. 2. Scale this vector by the scale factor. 3. Add the new vector back to the centre's coordinates.