Combined Transformation Worked Example

Why Combined Transformations Are Worth Mastering

Transformation questions appear on virtually every IGCSE Mathematics paper. On the Core syllabus, you need to perform and describe single transformations. On the Extended syllabus, you must also combine transformations and describe the result as a single transformation. These questions test spatial reasoning, coordinate skills, and your ability to communicate mathematically — all in one question.

Combined transformation questions are often worth five to eight marks, making them some of the highest-value questions on the paper. The key to success is precision: every transformation must be described using specific mathematical language, and even small errors in coordinates can cascade through the entire problem.

The Four IGCSE Transformations

Let us briefly review what each transformation requires in its description:

Translation: A column vector showing the horizontal and vertical movement
Reflection: The equation of the mirror line
Rotation: The centre of rotation, the angle, and the direction (clockwise or anticlockwise)
Enlargement: The centre of enlargement and the scale factor

Missing any one of these elements loses you a mark. This is the most common way students drop marks on transformation questions — giving an incomplete description.

The Problem

Triangle A has vertices at (1, 1), (3, 1), and (1, 4).

(a) Reflect triangle A in the line y = x. Label the image triangle B.

(b) Rotate triangle B by 90° clockwise about the origin. Label the image triangle C.

(c) Describe fully the single transformation that maps triangle A onto triangle C.

Part (a): Reflection in y = x

When you reflect a point in the line y = x, the x and y coordinates swap. This is a rule worth memorising:

(1, 1) → (1, 1) — this point lies on the mirror line, so it stays put
(3, 1) → (1, 3)
(1, 4) → (4, 1)

Triangle B has vertices at (1, 1), (1, 3), and (4, 1).

To verify, check that each original point and its image are equidistant from the line y = x, measured perpendicular to the line. For (3, 1), the distance to y = x is |3 − 1|/√2, and the distance from (1, 3) to y = x is |3 − 1|/√2. They match, confirming the reflection is correct.

Part (b): 90° Clockwise Rotation About the Origin

For a 90° clockwise rotation about the origin, the rule is: (x, y) → (y, −x). Apply this to each vertex of triangle B:

(1, 1) → (1, −1)
(1, 3) → (3, −1)
(4, 1) → (1, −4)

Triangle C has vertices at (1, −1), (3, −1), and (1, −4).

Part (c): Describing the Single Transformation

Now we need to find the single transformation that maps triangle A directly onto triangle C.

Triangle A: (1, 1), (3, 1), (1, 4) Triangle C: (1, −1), (3, −1), (1, −4)

Looking at the coordinates:

(1, 1) → (1, −1): the x-coordinate stays the same, the y-coordinate is negated
(3, 1) → (3, −1): same pattern
(1, 4) → (1, −4): same pattern

Every point has been mapped from (x, y) to (x, −y). This is a reflection in the x-axis.

The complete description: “Reflection in the line y = 0” (or equivalently, “Reflection in the x-axis”).

Why the Description Must Be Complete

If you wrote only “reflection” without specifying the mirror line, you would lose a mark. The examiners are very strict about this. Here is what a complete description looks like for each transformation type:

Translation: “Translation by the vector (3, −2)” — you must give the vector
Reflection: “Reflection in the line y = −x” — you must give the mirror line equation
Rotation: “Rotation of 90° anticlockwise about the point (2, 1)” — you must give the angle, direction, and centre
Enlargement: “Enlargement with scale factor 2, centre (0, 0)” — you must give the scale factor and centre

How to Identify the Transformation Type

When looking at the original and image shapes to identify the single transformation:

Same size, same orientation: It is a translation. Find the column vector.
Same size, different orientation (flipped): It is a reflection. Find the mirror line by connecting corresponding points and finding the perpendicular bisector.
Same size, rotated: It is a rotation. Find the centre by using perpendicular bisectors of lines joining corresponding points.
Different size: It is an enlargement. Find the scale factor by comparing corresponding lengths. Find the centre by drawing lines through corresponding points — they intersect at the centre.

Working With Coordinate Rules

Memorising these coordinate rules saves enormous time on the exam:

Reflection in x-axis: (x, y) → (x, −y)
Reflection in y-axis: (x, y) → (−x, y)
Reflection in y = x: (x, y) → (y, x)
Reflection in y = −x: (x, y) → (−y, −x)
90° clockwise about origin: (x, y) → (y, −x)
90° anticlockwise about origin: (x, y) → (−y, x)
180° rotation about origin: (x, y) → (−x, −y)

These rules only work when the centre of rotation is the origin or the mirror line passes through the origin. For other centres or mirror lines, you need to work point by point using the geometric definitions.

Common Mistakes

Confusing clockwise and anticlockwise: On a standard set of axes, anticlockwise is the “positive” direction. Double-check which way the question asks.
Applying transformations in the wrong order: The order matters. Reflecting then rotating gives a different result from rotating then reflecting.
Incomplete descriptions: As noted above, every transformation type requires specific pieces of information. Missing any one loses a mark.
Plotting errors: When working on graph paper, plot each transformed vertex carefully and double-check before drawing the image shape.
Forgetting that enlargement can have a negative scale factor: A negative scale factor means the image is on the opposite side of the centre and inverted.

Practice Questions

Reflect the triangle with vertices (2, 3), (5, 3), (2, 7) in the y-axis, then rotate the image 180° about the origin. Describe the single transformation.
Enlarge a shape with scale factor 2, centre (1, 1), then translate by vector (3, 0). Can this be described as a single transformation?
Rotate the point (4, 2) by 90° anticlockwise about the point (1, 1). What are the coordinates of the image?

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