Transformation Geometry — Grade 8 & 9 Maths (CAPS)

Every transformation an exam sets — reflection in the axes and y = x, translation, rotation about the origin, and enlargement — with the coordinate rule for each, worked examples and a worksheet.

Also known as: translations, reflections and rotations · enlargement · dilation

A transformation changes the position (or size) of a shape on the Cartesian plane. Three of them — reflection, translation and rotation — keep the shape exactly the same size (the image is congruent); a fourth, enlargement, resizes it. Each has a coordinate rule you can apply to any point.

1The coordinate rules (learn these)

Reflect in the x-axis: (x ; y) → (x ; −y).
Reflect in the y-axis: (x ; y) → (−x ; y).
Reflect in the line y = x: (x ; y) → (y ; x).
Translation: add the shift to each coordinate, e.g. 3 right, 2 up → (x + 3 ; y + 2).
Rotate 180° about the origin: (x ; y) → (−x ; −y).
Enlargement by factor k from the origin: (x ; y) → (kx ; ky).

Reflecting a point in the x-axis: the x stays, the y flips sign. (3 ; 5) → (3 ; −5).

2Type 1: Reflection

Worked ExampleExample 1: reflect (3 ; 5) in the x-axis

1
Rule: (x ; y) → (x ; −y).
Reflecting in the x-axis keeps x, flips the sign of y.
2
So (3 ; 5) → (3 ; −5).
The point drops to the mirror position below the axis.

Worked ExampleExample 2: reflect (4 ; −2) in the line y = x

1
Rule: (x ; y) → (y ; x).
Reflecting in y = x swaps the coordinates.
2
So (4 ; −2) → (−2 ; 4).
Just swap the two numbers.

3Type 2: Translation

Worked ExampleExample 3: translate (2 ; 3) by 4 right and 1 down

1
4 right adds 4 to x; 1 down subtracts 1 from y.
Right/up are positive, left/down negative.
2
(2 + 4 ; 3 − 1) = (6 ; 2).
Apply the shift to each coordinate.

4Type 3: Rotation about the origin

Worked ExampleExample 4: rotate (3 ; 5) by 180° about the origin

1
Rule: (x ; y) → (−x ; −y).
A half-turn flips the sign of both coordinates.
2
So (3 ; 5) → (−3 ; −5).
The point moves to the opposite side of the origin.

5Type 4: Enlargement

An enlargement multiplies every coordinate by a scale factor k (from the origin). A factor greater than 1 enlarges; between 0 and 1 reduces.

Worked ExampleExample 5: enlarge (3 ; 4) by scale factor 2

1
Multiply each coordinate by 2: (2 × 3 ; 2 × 4).
Enlargement rule (x ; y) → (kx ; ky).
2
= (6 ; 8).
The shape is now twice as far from the origin in every direction.
3
The AREA becomes 2² = 4 times as big.
Area scales by the factor squared.

💡

Reflection, translation and rotation don't change size or shape — the image is congruent to the original. Only enlargement changes the size.

⚠️

For enlargement, area scales by the factor squared. Doubling the sides (factor 2) makes the area 4 times as big, not 2 times.

Transformations reappear in Grade 10–12 function graphs (shifts and reflections of curves), so the coordinate rules here are worth knowing cold. Practise every type below.

6Lines of symmetry

A line of symmetry is a line you could fold a shape along so that the two halves land exactly on top of each other. It is a reflection where the shape maps onto itself.

A rectangle has 2 lines of symmetry, an equilateral triangle 3, and a parallelogram none at all.

Rectangle: 2 — one horizontal, one vertical. The diagonals are not lines of symmetry.
Square: 4 — the rectangle's two, plus both diagonals.
Equilateral triangle: 3 — one from each vertex to the midpoint of the opposite side.
Isosceles triangle: 1. Scalene triangle: 0.
Rhombus: 2 (the diagonals). Kite: 1. Parallelogram: 0.
Regular polygon with n sides: n. Circle: infinitely many.

⚠️

Claiming a rectangle's diagonal is a line of symmetry. Fold a non-square rectangle along its diagonal and the halves do not match — the two triangles are congruent but they land the wrong way round. Only in a square does the diagonal work.

💡

A parallelogram catches people out because it looks balanced. It has rotational symmetry — turn it 180° and it fits itself — but no line you can fold it along. Balance is not the same as reflective symmetry.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Reflect (2 ; 7) in the y-axis. (reflect)
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(−2 ; 7)
2
Reflect (4 ; −1) in the x-axis. (reflect)
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(4 ; 1)
3
Reflect (5 ; 2) in the line y = x. (reflect)
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(2 ; 5)
4
Reflect (−3 ; 6) in the y-axis. (reflect)
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(3 ; 6)
5
Translate (1 ; 4) by 5 right and 2 up. (translate)
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(1 + 5 ; 4 + 2) = (6 ; 6)
6
Translate (3 ; −2) by 2 left and 3 up. (translate)
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(3 − 2 ; −2 + 3) = (1 ; 1)
7
Rotate (2 ; 5) by 180° about the origin. (rotate)
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(−2 ; −5)
8
Rotate (−4 ; 1) by 180° about the origin. (rotate)
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(4 ; −1)
9
Enlarge (3 ; 4) by scale factor 2 from the origin. (enlarge)
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(6 ; 8)
10
Enlarge (4 ; 6) by scale factor 1/2 from the origin. (reduce)
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(2 ; 3)
11
Write down the number of lines of symmetry of a square, a rectangle and a parallelogram.
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Square 4 ✓; rectangle 2 ✓; parallelogram 0 ✓
12
Explain why a diagonal of a rectangle is NOT a line of symmetry.
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Folding along the diagonal does not make the two halves coincide ✓ — the halves are congruent but do not land on top of each other, unless the rectangle is a square ✓
13
How many lines of symmetry does a regular hexagon have, and how many does an isosceles triangle have?
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Regular hexagon 6 ✓; isosceles triangle 1 ✓
14
A parallelogram has no lines of symmetry but still looks balanced. Explain.
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It has rotational symmetry — turning it 180° about its centre maps it onto itself ✓ — but there is no line it can be folded along so the halves match ✓