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On this MathsLinks page you can download free printable invariant points PDF worksheets (Qs) and full worked solutions or mark schemes (Ans) from Ken Stafford, Corbett Textbook, Maths Made Easy, E Classroom, PMT, plus a printable one-page invariant points revision guide.

What GCSE grade is invariant points (8M)?

Invariant points (topic code 8M) is a Grade 8+ geometry topic in GCSE Maths for Edexcel, AQA, OCR, Eduqas and IGCSE specifications.

Free Invariant points Worksheets (PDF) with Answers — GCSE Maths

Free Invariant points PDF Worksheets & Answers

Grade 8+ • Topic 8M

Resource	Questions	Answers
Ken Stafford GCSE past paper questions & solutions	Qs	Ans
Corbett Textbook Textbook Exercise & answers	Qs	Ans
Maths Made Easy MME worksheet & model answers	Qs	Ans
E Classroom GCSE topic booklet & answers	Qs	Ans
PMT (Edxl) Edexcel GCSE topic questions & solutions	Qs	Ans
PMT (AQA) AQA GCSE topic questions & solutions	Qs	Ans
PMT (IGCSE) Edexcel IGCSE topic questions & solutions	Qs	Ans

Grade 8+ topics
8A - Quad. sim. equations	8B - Transforming graphs y=f(x)
8C - Algebraic proof	8D - Completing the square
8E - Quadratic inequalities	8F - Quadratic sequences
8G - Velocity time graphs	8H - Circles and tangents
8J - Circle theorem proof	8K - Vector proof
8L - Probability equation questions	8M - Invariant points
8N - Spheres, cones and cylinders	8P - Volume of a frustum

Explanation:

An invariant point is a point on a shape that does not move (maps onto itself) after a single transformation or a combination of transformations.

• Under a reflection, every point lying directly on the mirror line is invariant.
• Under a rotation (not 360°) or enlargement (scale factor k ≠ 1), only the centre of rotation or enlargement is invariant.
• Under a non-zero translation, there are no invariant points. For combined transformations, test vertices systematically or track (x, y) algebraically.

Example 1: Triangle ABC has vertices A(1, 3), B(4, 4), and C(4, 1). Which vertex is invariant under a reflection in the line y = x?

Solution:

Points on y = x stay fixed: A(1,3) → (3,1); C(4,1) → (1,4)

B(4, 4) lies on y = x ⇒ Answer: (4, 4)

Example 2: A point (x, y) is reflected in the x-axis and then translated by vector 06. Find the equation of the line of invariant points.

Solution:

Reflection in x-axis maps (x, y) → (x, −y)

Translation by 06 maps (x, −y) → (x, −y + 6)

For invariance, equate y-coordinates: −y + 6 = y

2y = 6 ⇒ y = 3 (for any x)

Answer: y = 3

Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Invariant points Questions & Answers

10-Question Interactive Quiz More Past Paper Questions

Past Paper Example 1 AQA Jun 2017 2H Q26 2 marks

Full Paper (PDF) • Mark Scheme (PDF)

(a) A sketch of a quadrilateral ABCD is shown.
ABCD is enlarged, centre B, scale factor 13.
Circle the vertex that is invariant.
A B C D
26 (b) A sketch of a quadrilateral PQRS is shown.
PQRS is reflected in the line y = x.
Circle the vertex that is invariant.
P Q R S

(a) A sketch of a quadrilateral ABCD is shown.
ABCD is enlarged, centre B, scale factor 13.
Circle the vertex that is invariant.
A B C D [1 mark]

(b) A sketch of a quadrilateral PQRS is shown.
PQRS is reflected in the line y = x.
Circle the vertex that is invariant.
P Q R S [1 mark]

Worked Solution & Mark Scheme (2 marks)

Part (a) Solution:

The centre of an enlargement remains invariant (it does not move)
Since the centre is B, vertex B is invariant. B1

Answer (a): B

Part (b) Solution:

Under reflection in the line y = x, points lying on the line y = x are invariant
For vertex P(1, 1), the x and y coordinates are equal, so P lies on y = x and is invariant. B1

Answer (b): P

Past Paper Example 2 AQA Nov 2020 2H Q24 3 marks

Full Paper (PDF) • Mark Scheme (PDF)

(a) Quadrilateral ABCD is shown.
Work out the coordinates of C when ABCD is
rotated 90° clockwise about O
then
translated by (-62)
24 (b) Triangle PQR is shown.
When PQR is reflected in a line, P and R are invariant points.
Circle the equation of the line.
y = x + 6 y = -x y = 2 x = -4

(a) Quadrilateral ABCD is shown.
Work out the coordinates of C when ABCD is
rotated 90° clockwise about O
then
translated by (-62) [2 marks]

(b) Triangle PQR is shown.
When PQR is reflected in a line, P and R are invariant points.
Circle the equation of the line.
y = x + 6 y = -x y = 2 x = -4 [1 mark]

Worked Solution & Mark Scheme (3 marks)

Part (a) Solution:

Point C has coordinates (4, 1) M1
Rotating 90° clockwise about (0,0) maps (x, y) → (y, -x), so (4, 1) → (1, -4)
Translating by (-62) gives (1 - 6, -4 + 2) = (-5, -2). A1

Answer (a): Answer (-5-2)

Part (b) Solution:

Invariant points must lie on the mirror line
The line through P(-4, 4) and R(-2, 2) has gradient 2 - 4-2 - (-4) = -1 and passes through (0,0), so y = -x. B1

Answer (b): y = y = -x

Past Paper Example 3 GCSE Nov 2020 3H Q11 4 marks

Full Paper (PDF) • Mark Scheme (PDF)

The diagram shows a triangle **P** on a grid.

(a) Describe fully the single transformation that maps triangle **P** onto triangle **R**. [3 marks]

(b) Write down the coordinates of point A. [1 mark]

Worked Solution & Mark Scheme (4 marks)

Part (a) Solution:

P vertices: (1,2), (1,5), (2,5) → Q: (–1,–2), (–1,–5), (–2,–5) → R: (4,–4), (4,–7), (3,–7) M2 C1

Answer (a): Rotation of 180° about (2.5, –1)

Part (b) Solution:

Midpoint of (1,2) and (4,–4) is (2.5, –1). B1

Answer (b): (2.5, –1)

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Invariant points

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GCSE Past Paper Invariant points Questions & Answers