Invariant points Quiz (Grade 8+)

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Invariant points — Notes & Worked Examples Show / Hide

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

One Page Guide (PDF) Invariant points Worksheets

Question 1 AQA Nov 2018 3H Q25

[1 mark]

ABCD is a quadrilateral.
The quadrilateral is reflected in the line x = 4

Which vertices are invariant?
Circle your answer.
A and D C and D B and C B and D

Worked Solution & Answer

Points that lie on the mirror line x = 4 do not move under reflection
Both B (4, 1) and C (4, 4) have x = 4, so vertices B and C are invariant. B1

Answer: B and C

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Question 2

[2 marks]

Triangle A has vertices (–4, 1), (–3, 1), (–3.5, 4). It is rotated 180° about (–2, 0) followed by translation by vector (–3, 2). Find the coordinates of the invariant point on triangle A.

Triangle **A** is transformed by the combined transformation of a rotation of 180° about the point (–2, 0) followed by a translation with vector (-32)
One point on triangle **A** is invariant under the combined transformation.
Find the coordinates of this point.

Worked Solution & Answer

Rotation of 180° about (–2, 0) maps (x, y) to (–4 – x, –y) M1
Translation by (–3, 2) maps this to (–7 – x, 2 – y)
For invariance: x = –7 – x ⇒ x = –3.5, and y = 2 – y ⇒ y = 1. Thus (–3.5, 1). A1

Answer: –3.5, 1

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Question 3

[2 marks]

Square ABCD is transformed by a combined transformation of a reflection in the line x = -1 followed by a rotation.
Under the combined transformation, two vertices of the square ABCD are invariant.
Describe fully one possible rotation.

Worked Solution & Answer

Reflecting ABCD in x = –1 gives A'(–3, 4), B'(–5, 4), C'(–5, 2), D'(–3, 2) M1
Rotating 90° clockwise about (–1, 0) maps B'(–5, 4) → B(3, 4) and D'(–3, 2) → D(1, 2), leaving B and D invariant. C1

Answer: Rotation 90° clockwise about (–1, 0) [or 90° anticlockwise about (–1, 6), or 180° about (–1, 2), or 180° about (–1, 4)]

Question Paper (PDF) • Mark Scheme (PDF)

Question 4 AQA Jun 2017 2H Q26

[2 marks]

(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 & Answer

(a)

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

Answer (a): B

(b)

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

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Question 5 AQA Nov 2020 2H Q24

[3 marks]

(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]

Answer

(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]

y =

Worked Solution & Answer

(a)

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)

(b)

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

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Question 6

[3 marks]

(a) Enlarge triangle P by scale factor –1.5 with centre (–2, –1). Label your image Q. (2) (b) Triangle P is rotated 90° anticlockwise about O then translated so 1 vertex is invariant. Find one possible column vector. (1)

(a) Enlarge triangle **P** by scale factor -112 with centre of enlargement (-2, -1)
Label your image **Q**. [2 marks]

(b) Find one possible column vector for the translation. [1 mark]

Worked Solution & Answer

(a)

Vectors from (–2, –1) to (–8, 1), (–6, 1), (–8, 5) are (–6, 2), (–4, 2), (–6, 6) M1
multiply by –1.5 to get (9, –3), (6, –3), (9, –9) ⇒ (7, –4), (4, –4), (7, –10) A1

Answer (a): Triangle Q with vertices (7, –4), (4, –4), (7, –10)

(b)

90° anticlockwise sends (x, y) → (–y, x): (–8, 1) → (–1, –8), translation = (–7, 9)
(–6, 1) → (–1, –6), translation = (–5, 7)
(–8, 5) → (–5, –8), translation = (–3, 13) B1

Answer (b): –7, 9

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Question 7 OCR Jun 2024 5H Q19

[3 marks]

(a) Describe fully the **single** transformation that is equivalent to:
• a rotation of 20° anticlockwise about the origin, followed by
• a rotation of 70° clockwise about the origin.
(b) The diagram shows the coordinates of the vertices of a rectangle.

(a) Describe fully the **single** transformation that is equivalent to:
• a rotation of 20° anticlockwise about the origin, followed by
• a rotation of 70° clockwise about the origin. [2 marks]

(b) The diagram shows the coordinates of the vertices of a rectangle. [1 mark]

The rectangle is reflected in the line y = 4.
Write down the coordinates of the vertices of the rectangle that are invariant.

Worked Solution & Answer

(a)

70° clockwise − 20° anticlockwise = 50° clockwise about (0, 0). M1 C1

Answer (a): Rotation of 50° clockwise about the origin (0, 0).

(b)

Points on the mirror line y = 4 are invariant: (1, 4) and (5, 4). B1

Answer (b): (1, 4) and (5, 4)

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Question 8 AQA Nov 2022 1H Q24

[4 marks]

Triangle ABC is drawn on a grid.

ABC is transformed to A'B'C' by a reflection in the line x = 1
A'B'C' is transformed to A''B''C'' by a rotation 90° anticlockwise about (1, -4)
Which **one** point on ABC is invariant under the combined transformation?
You **must** show the result of each transformation on the grid.

Worked Solution & Answer

Reflecting A(–4, 3), B(–4, 1), C(–1, 1) in x = 1 gives A'(6, 3), B'(6, 1), C'(3, 1) M2
Rotating 90° anticlockwise about (1, –4) gives A''(–6, 1), B''(–4, 1), C''(–4, –2) M1
Point B (–4, 1) is invariant. A1

Answer: Answer (-4, 1)

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Question 9

[4 marks]

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 & Answer

(a)

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)

(b)

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

Answer (b): (2.5, –1)

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Question 10 AQA Jun 2019 3H Q25

[5 marks]

ABCD is a square.
A is (–2, 1), B is (0, –1), C is (2, 1), D is (0, 3)

(a) A single transformation of ABCD is such that
B is mapped to D
D is mapped to B
A and C are invariant points.
Describe fully the transformation. [2 marks]

(b) A different single transformation of ABCD is such that
B is mapped to D
D is mapped to B
the only invariant point is (0, 1)
Describe fully the transformation. [3 marks]

Worked Solution & Answer

(a)

The invariant line passing through A(–2, 1) and C(2, 1) is the horizontal line y = 1 M1
Reflecting in y = 1 swaps B(0, –1) and D(0, 3). A1

Answer (a): Answer Reflection in the line y = 1

(b)

The centre of the square is (0, 1) M1
A rotation of 180° about (0, 1) maps B(0, –1) to D(0, 3) and D(0, 3) to B(0, –1) M1
The only fixed point is the centre of rotation (0, 1). A1

Answer (b): Answer Rotation of 180° about (0, 1)

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Score: 0 / 10 (0 / 29 marks)

Invariant points Worksheets Next: Spheres, cones and cylinders →

Grade 8+ Quizzes

14 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