Mathematics questions, chapter 11

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Isometric Transformations

42 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working for Section B.

Section A: objective questions

12 marks

Choose the correct answer. Each question carries 1 mark.

Question 1

Which of the following can change when an object undergoes a translation, a reflection or a rotation?

The size of the object
The shape of the object
The position of the object
The length of its sides

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An isometric transformation moves the object, so its position changes. The shape, size and lengths stay the same, so the image is congruent to the object.

For the teacher: Remembering, low difficulty, standard 11.1.1

Textbook p. 209

Question 2

Two figures are congruent if they have

the same shape and the same size, whatever their orientation
the same shape only, whatever their size
the same size and the same orientation only
the same position on the plane

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Congruent figures have the same shape and size. Position and orientation may be different. Figures with the same shape but different sizes are similar.

For the teacher: Remembering, low difficulty, standard 11.1.2

Textbook p. 210

Question 3

What does the translation −32 mean?

3 units to the right and 2 units upwards
3 units to the right and 2 units downwards
2 units to the left and 3 units upwards
3 units to the left and 2 units upwards

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The top number is the movement along the x-axis: −3 is 3 units to the left. The bottom number is along the y-axis: 2 is 2 units upwards.

For the teacher: Understanding, low difficulty, standard 11.2.2

Textbook p. 214

Question 4

Find the image of the point (5, −3) under the translation −24.

(7, 1)
(3, 1)
(3, −7)
(7, −7)

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(5 + (−2), −3 + 4) = (3, 1). (7, 1) adds −2 as +2. (3, −7) subtracts the 4 instead of adding.

For the teacher: Applying, low difficulty, standard 11.2.3

Textbook p. 217

Question 5

Diagram 1 shows the point M and its image M′ under a translation. What is the translation?

Diagram 1

6−4
−64
26
64

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Vector = image − object = (4 − (−2), 1 − 5) = (6, −4). The image is 6 units to the right and 4 units down. −64 is the reverse translation.

For the teacher: Applying, low difficulty, standard 11.2.3

Textbook p. 217

Question 6

Which statement about a reflection is not true?

The object and its image are the same distance from the axis of reflection.
The image has the same size as the object.
A point on the axis of reflection is mapped onto itself.
The image has the same orientation as the object.

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In a reflection the orientation is reversed (a mirror image). It is a translation or a rotation that keeps the orientation.

For the teacher: Understanding, low difficulty, standard 11.3.1

Textbook p. 219

Question 7

What is the image of the point (4, −7) under a reflection in the y-axis?

(4, 7)
(7, −4)
(−4, 7)
(−4, −7)

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Reflection in the y-axis changes the sign of x only: (x, y) becomes (−x, y), so (4, −7) becomes (−4, −7). (4, 7) is the reflection in the x-axis.

For the teacher: Applying, low difficulty, standard 11.3.3

Textbook p. 221

Question 8

The point L(4, 1) is mapped to L′(4, 5) under a reflection. What is the equation of the axis of reflection?

y = 2
y = 3
y = 4
x = 4

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The x-coordinate does not change, so the axis is horizontal. It passes through the midpoint of 1 and 5, which is (1 + 5) ÷ 2 = 3. The axis is y = 3.

For the teacher: Analysing, medium difficulty, standard 11.3.4

Textbook p. 221

Question 9

A complete description of a rotation must state

the vector and the angle
the axis, the angle and the size
the centre, the angle and the direction
the object, the image and the axis

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A rotation is described by its centre of rotation, the angle and the direction (clockwise or anticlockwise).

For the teacher: Remembering, low difficulty, standard 11.4.2

Textbook p. 224

Question 10

What is the image of the point (3, 2) under a rotation of 90° anticlockwise about the origin?

(−2, 3)
(2, −3)
(−3, −2)
(2, 3)

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For 90° anticlockwise about the origin, (x, y) becomes (−y, x), so (3, 2) becomes (−2, 3). (2, −3) is the 90° clockwise image and (−3, −2) is the 180° image.

For the teacher: Applying, low difficulty, standard 11.4.3

Textbook p. 225

Question 11

Which statement describes an isometry?

Every length of the object is doubled.
The distance between any two points of the object is kept in the image.
The shape of the object changes but its area stays the same.
The image is smaller than the object.

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An isometry keeps the distance between any two points, so the image is congruent to the object. Changing the size or shape is not an isometry.

For the teacher: Understanding, low difficulty, standard 11.5.2

Textbook p. 231

Question 12

Diagram 2 shows a propeller with three identical blades. What is the order of rotational symmetry of the propeller about its centre?

Propeller with three identical blades spaced 120 degrees apart about a central dot — Diagram 2

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The three blades are 120° apart, so the propeller fits itself after 120°, 240° and 360°. The order is 360° ÷ 120° = 3.

For the teacher: Understanding, low difficulty, standard 11.6.2

Textbook p. 235

Section B: structured questions

30 marks

Answer all questions. Show your working.

Question 13

Diagram 3 shows triangle PQR on a Cartesian plane. The triangle is translated by −32.

Triangle PQR with P(2, 1), Q(5, 2) and R(2, 4) — Diagram 3

[2 marks] (a)State the coordinates of the images P′, Q′ and R′.

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P′ = (−1, 3), Q′ = (2, 4) [1]1
R′ = (−1, 6) [1]1

Tick each point you got.

[1 mark] (b)Point S′(4, 1) is the image of S under the translation −52. Find the coordinates of S.

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S = (4 + 5, 1 − 2) = (9, −1) [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 11.2.3

Textbook p. 216

Question 14

Diagram 4 shows triangle ABC and the line x = −1. Triangle ABC is reflected in the line x = −1.

Triangle ABC with A(1, 4), B(3, 1), C(1, 1) and the vertical line x = −1 — Diagram 4

[2 marks] (a)State the coordinates of A′, B′ and C′.

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A and C are 2 units from the line, so A′ and C′ have x = −3 [1]1
A′ = (−3, 4), B′ = (−5, 1), C′ = (−3, 1) [1]1

x′ = 2 × (−1) − x. For B: −2 − 3 = −5.

Tick each point you got.

[1 mark] (b)State the image of the point (−1, 6).

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(−1, 6) [1]1

The point is on the axis of reflection, so it does not move.

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 11.3.3

Textbook p. 220

Question 15

Diagram 5 shows triangle ABC. It is rotated through 90° clockwise about the origin O.

Triangle ABC with A(2, 1), B(5, 1) and C(5, 3) in the first quadrant — Diagram 5

[3 marks] (a)State the coordinates of A′, B′ and C′.

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Rule for 90° clockwise: (x, y) becomes (y, −x) [1]1
A′ = (1, −2), B′ = (1, −5) [1]1
C′ = (3, −5) [1]1

Tick each point you got.

[2 marks] (b)The image of a point T is (−2, −3) under a rotation of 90° anticlockwise about the origin. Find the coordinates of T.

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Rotate (−2, −3) through 90° clockwise [1]1
T = (−3, 2) [1]1

Check: rotating (−3, 2) 90° anticlockwise gives (−2, −3).

Tick each point you got.

For the teacher: Applying, low difficulty, standard 11.4.3

Textbook p. 228

Question 16

In each part the centre of rotation is not the origin O.

[2 marks] (a)Find the image of M(2, 5) under a rotation of 90° anticlockwise about N(1, 2).

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Vector from N to M = (1, 3); after 90° anticlockwise = (−3, 1) [1]1
M′ = (1 − 3, 2 + 1) = (−2, 3) [1]1

Tick each point you got.

[2 marks] (b)K′(3, −1) is the image of K under a rotation of 90° clockwise about (1, 1). Find K.

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Rotate K′ through 90° anticlockwise about (1, 1): vector (2, −2) becomes (2, 2) [1]1
K = (3, 3) [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 11.4.3

Textbook p. 229

Question 17

In Diagram 6, triangles A, B and C are images of triangle P under three different isometries.

Triangle P with vertices (1, 1), (4, 1), (1, 3); A is the same triangle 5 units to the right; B is its mirror image below the x-axis; C is a quarter-turn image in the second quadrant — Diagram 6

[3 marks] (a)State the isometry that maps P to A, to B and to C.

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A: translation 50 [1]1
B: reflection in the x-axis [1]1
C: rotation of 90° anticlockwise about the origin [1]1

Tick each point you got.

[1 mark] (b)Explain why P, A, B and C are all congruent.

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Translation, reflection and rotation are isometries, so the shape and size stay the same [1]1

Tick each point you got.

For the teacher: Analysing, medium difficulty, standard 11.5.1

Textbook p. 230

Question 18

State the order of rotational symmetry of each shape.

[1 mark] (a)A square

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4 [1]1

Tick each point you got.

[1 mark] (b)A rectangle that is not a square

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2 [1]1

Tick each point you got.

[1 mark] (c)A triangle with three sides of different lengths

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1 [1]1

Order 1 means no rotational symmetry.

Tick each point you got.

For the teacher: Applying, low difficulty, standard 11.6.2

Textbook p. 235

Question 19

Diagram 7 shows triangle P and its image P′.

Triangle P with vertices (1, 1), (3, 1), (1, 2) and triangle P′ with vertices (−1, −1), (−3, −1), (−1, −2) — Diagram 7

[1 mark] (a)Describe one single transformation that maps P onto P′.

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Rotation of 180° about the origin [1]1

Tick each point you got.

[2 marks] (b)Describe a different pair of transformations, one after the other, that also maps P onto P′.

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Reflect P in the x-axis, then reflect the result in the y-axis.

Reflection in the x-axis [1]1
followed by reflection in the y-axis (either order) [1]1

Any other correct pair is also accepted, for example two rotations of 90° about the origin in the same direction.

Tick each point you got.

For the teacher: Creating, medium difficulty, standard 11.5.3

Textbook p. 241

Question 20

The point P(2, 1) is mapped in this order: first a translation 3−2, then a reflection in the y-axis, then a rotation of 90° anticlockwise about the origin.

[3 marks] (a)Find the final image of P.

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After translation: (5, −1) [1]1
After reflection in the y-axis: (−5, −1) [1]1
After rotation: (1, −5) [1]1

Tick each point you got.

[2 marks] (b)Aini says the final image is the same if the reflection is done first. Is she correct? Show your working.

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Reflection first: (−2, 1), then translation: (1, −1), then rotation: (1, 1) [1]1
(1, 1) is not (1, −5), so she is not correct; the order matters [1]1

Tick each point you got.

For the teacher: Evaluating, high difficulty, standard 11.5.3

Textbook p. 241

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