Congruency, Enlargement and Combined Transformations
How to Identify a single transformation
Use this to name and fully describe the transformation that maps one shape onto another.
Before you start
- The four transformations: translation, reflection, rotation, enlargement
- Reading coordinates from a grid
- Comparing the side lengths of two shapes
When to use it
Use this to name and fully describe the transformation that maps one shape onto another.
The steps
- Compare the object and image: same size means an isometry, larger or smaller means an enlargement.
- If sizes match, decide between translation, reflection and rotation from the orientation.
- Find the specific details: vector, axis, or centre and angle.
- State the type and every detail in one full description.
- Check by applying your description to the object.
Worked example
Triangle A has vertices (1, 1), (1, 3) and (2, 1). Triangle B has vertices (4, 1), (4, 3) and (5, 1).
Describe fully the single transformation that maps A onto B.
- Both triangles have the same shape and size (legs 2 and 1), so it is an isometry, not an enlargement.
- The orientation is unchanged, B is A slid across with no turn or flip, so it is a translation.
- Take one vertex: (1, 1) → (4, 1), a change of +3 in x and 0 in y, so the vector is (3, 0).
- Full description: a translation by the column vector (3, 0), i.e. 3 units to the right.
- Check another vertex: (1, 3) + (3, 0) = (4, 3), which matches B. ✓
A second example, with a twist
The shapes are the same size but turned, so it is a rotation and you must find both the centre and the angle. Triangle P has vertices (1, 1), (4, 1) and (1, 2).
Triangle Q has vertices (−1, 1), (−1, 4) and (−2, 1). Describe fully the single transformation that maps P onto Q.
- Both triangles are the same size, so it is an isometry, not an enlargement.
- The orientation has turned (P points right, Q points up), so it is a rotation, not a translation or reflection.
- Test the origin as centre: a 90° anticlockwise turn sends (x, y) → (−y, x). Check (1, 1) → (−1, 1) ✓ and (4, 1) → (−1, 4) ✓.
- Full description: a rotation of 90° anticlockwise about the centre (0, 0).
- Check the last vertex: (1, 2) → (−2, 1), which matches Q. ✓
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
How do I tell a rotation from a reflection when both change orientation?
A reflection flips the shape, so it reads back-to-front like a mirror image; a rotation only turns it and keeps the same handedness. Trace the vertices in order: if their clockwise or anticlockwise sense reverses it is a reflection, but if the sense stays the same it is a rotation.
Do I lose marks if I forget one detail in the description?
Yes. Each transformation needs its full set of details: a translation needs the vector, a reflection needs the axis equation, a rotation needs centre, angle and direction, and an enlargement needs centre and scale factor.
Missing any one of these usually costs a mark, even when the type is correct.
How do I find the centre of a rotation?
Join each object point to its image and draw the perpendicular bisector of each of those lines. The point where the bisectors cross is the centre.
In the exam you can often test a likely centre such as the origin, then confirm the angle by checking that one vertex maps correctly.