Congruency, Enlargement and Combined Transformations

This chapter is about what happens to a shape when you move or resize it. You study congruency (same shape and size), enlargement (same shape, different size, with a scale factor), tessellation, and combined transformations where two moves are applied one after another.

Paper 2 gives shapes on a grid and asks you to describe a transformation fully, find an image, or work out an area using the scale factor. Full description marks are strict, a rotation needs centre, angle and direction, so precision is everything here.

The biggest source of lost marks in this chapter is an incomplete description, so learn exactly what each type must name. A translation is given by a vector, how far right or left and how far up or down, for example 3 units to the right and 2 units down.

A reflection needs the equation of the mirror line, such as x = 2, y = −1, y = x or y = −x. A rotation needs three separate details: the centre as coordinates, the angle (usually 90°, 180° or 270°) and the direction (clockwise or anticlockwise), only a 180° rotation lets you skip the direction.

An enlargement needs the centre and the scale factor. Writing 'a rotation about the origin' and stopping describes nothing the examiner can award, because each required detail carries its own mark.

A scale factor does not have to be a whole number bigger than 1. When it lies between 0 and 1, the image is smaller than the object, a reduction, though the syllabus still calls the transformation an enlargement.

When the scale factor is negative, the image lands on the opposite side of the centre and is turned upside down, which is the same as an enlargement by the positive size followed by a half-turn about the centre. For example, an enlargement of centre O(0, 0) and scale factor −2 sends the point (3, 1) to (−6, −2): each distance from the centre is still multiplied by 2, but the direction flips.

Area behaves the same way whatever the sign, because it is multiplied by the scale factor squared, and a square is never negative, a scale factor of −3 and a scale factor of +3 both multiply area by 9.

Two higher-mark tasks reward a little construction. To find the centre of an enlargement, draw a straight line through each object point and its matching image point; all these lines meet at one point, and that meeting point is the centre.

The scale factor is then the image distance divided by the object distance measured from that centre. The second task is the classic Paper 2 twist: a shape is transformed twice, and you are asked for the single transformation that has the same effect.

Track two or three key points from the original all the way to the final image, then ask which single move, translation, reflection, rotation or enlargement, carries the original there, and describe it with the full checklist. It is often a rotation or a translation even though two different steps were used to build it.

Here is a Paper 2-style question that combines an enlargement with a translation and asks for an area, exactly the kind where the scale factor must be squared.