Mathematics notes, chapter 11
Isometric Transformations
Textbook Mathematics Form 2 (DLP), pages 206–243
What this chapter covers
- Recognise a transformation and describe the change in position, direction and orientation; explain one-to-one correspondence between object and image (11.1.1)
- Explain congruence and identify congruent figures and corresponding sides and angles (11.1.2)
- Recognise a translation, describe it as a column vector, and find the image or the object, including problems with two moves (11.2.1 to 11.2.4)
- Recognise a reflection, describe it by its axis, find the image or object and the axis, and solve problems (11.3.1 to 11.3.4)
- Recognise a rotation, describe it by centre, angle and direction, find the image or object and the centre, and solve problems (11.4.1 to 11.4.3)
- Explain that translation, reflection and rotation are isometries and that an isometry gives a congruent image; solve problems on isometry and congruence (11.5.1 to 11.5.3)
- Explain rotational symmetry and find the order of rotational symmetry of a shape (11.6.1, 11.6.2)
Look at your reflection in a mirror, a ceiling fan turning and a person walking along a road. Each one is a transformation: the same shape is moved from one place to another. In this chapter you will learn three transformations, translation, reflection and rotation, and see why they give images that are congruent to the object.
| English | Bahasa Melayu |
|---|---|
| transformation, object, image | penjelmaan, objek, imej |
| translation, vector | translasi, vektor |
| reflection, axis of reflection | pantulan, paksi pantulan |
| rotation, centre of rotation | putaran, pusat putaran |
| clockwise, anticlockwise | ikut arah jam, lawan arah jam |
| isometry, congruence, orientation | isometri, kekongruenan, orientasi |
| rotational symmetry, order of rotational symmetry | simetri putaran, peringkat simetri putaran |
11.1Transformations and congruence
Transformation is the movement of every point of a figure on a plane by a fixed rule. The original figure is the object; the figure after the move is the image. Each point of the object matches exactly one point of the image (one-to-one correspondence).
A transformation can change the position, direction or orientation of a figure. In this chapter the shape and size stay the same, so the image is congruent to the object.
| Everyday action | Type of transformation |
|---|---|
| Akmal walks from his home to the bus stop | Translation (slides in a straight line) |
| Akmal sees himself in the mirror | Reflection (flips over a line) |
| A ceiling fan turns | Rotation (turns about a fixed point) |
Congruent figures have the same shape and the same size. Their position and orientation may be different. Corresponding sides are equal and corresponding angles are equal.
Example 1 (textbook p. 209): which is a transformation?
A figure that only moves, turns or flips is a transformation (shape unchanged). A figure whose appearance changes, for example a square that becomes a stretched rectangle, is not a transformation of this kind.Example 2 (textbook p. 210): matching object and image
Hexagon ABCDEF is mapped to PQRSTU, with C mapped to T.The image of point C is T. The image of line AB is the line RS, with the same length. The image of ∠BCD is ∠STU, with the same size.
A 10 sen coin and a 20 sen coin are both round but they are similar, not congruent, because their sizes are different. Congruent needs the same shape and the same size.
Triangle ABC has sides 6 cm, 8 cm and 10 cm. Triangle PQR has sides 10 cm, 6 cm and 8 cm. Are they congruent? Why?
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Yes. All three pairs of sides are equal, so the two triangles have the same shape and size (the order of the lengths does not matter).
11.2Translation
Translation moves every point of a figure the same distance in the same direction. The object and image have the same shape, size and orientation.
Vector of translation
A translation is written as a column vector ab:
- a is the movement parallel to the x-axis: positive to the right, negative to the left.
- b is the movement parallel to the y-axis: positive upwards, negative downwards.
| Description | Vector |
|---|---|
| 2 units to the left and 3 units downwards | −2−3 |
| 5 units to the right and 5 units downwards | 5−5 |
| 6 units downwards | 0−6 |
| 3 units to the right | 30 |
Finding the image or the object
Under a translation ab: object (x, y) → image (x + a, y + b).
To go back from the image to the object, subtract: object = (x′ − a, y′ − b).
To go back from the image to the object, subtract: object = (x′ − a, y′ − b).
Example 3: image
Find the image of Q(3, 1) under the translation −52.Q′ = (3 + (−5), 1 + 2) = (−2, 3)
Example 4: object
The image A′ is (−6, 1) under the translation 3−2. Find A.A = (−6 − 3, 1 − (−2)) = (−9, 3)
Example 5: vector from object and image
P(2, 9) is mapped to P′(3, 6).Vector = 3 − 26 − 9 = 1−3
Always write “image minus object”: (x′ − x, y′ − y). Check by adding the vector back to the object.
Do not swap the two numbers. The top number is the x-direction (left/right) and the bottom number is the y-direction (up/down).
(a) Find the image of (5, −3) under −24. (b) M(−2, 5) is mapped to M′(4, 1). Write the vector of translation.
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(a) (5 − 2, −3 + 4) = (3, 1)
(b) 4 − (−2)1 − 5 = 6−4
(b) 4 − (−2)1 − 5 = 6−4
Problems with two moves
For two translations one after the other, add the vectors. If a piece moves −40 then 3−2, the single translation is −4 + 30 + (−2) = −1−2.
11.3Reflection
Reflection flips every point of a figure over a line called the axis of reflection. The image is the mirror image of the object.
Properties of a reflection
- The object and image are on opposite sides of the axis.
- Each point and its image are the same perpendicular distance from the axis, so the axis is the perpendicular bisector of the line joining them.
- The shape and size are the same, but the orientation is reversed.
- A point on the axis is its own image (it does not move).
| Reflection of (x, y) | Image | Example: (4, −7) |
|---|---|---|
| in the x-axis | (x, −y) | (4, 7) |
| in the y-axis | (−x, y) | (−4, −7) |
Example 6: find the axis of reflection
M(−2, 1) is mapped to M′(−2, −1). The x-coordinate stays the same, so the axis is horizontal. The midpoint of 1 and −1 is 0, so the axis is the x-axis.For L(4, 1) mapped to L′(4, 5): midpoint of 1 and 5 is 3, so the axis is y = 3.
Example 7: reflection in a line x = k
Reflect A(1, 4) in the line x = −1. A is 1 − (−1) = 2 units to the right of the line, so A′ is 2 units to the left: x = −1 − 2 = −3. The y-coordinate does not change, so A′ = (−3, 4).Axis of reflection: take the midpoint of the object and image points. If the y-coordinates are equal, the axis is vertical, x = (x + x′) ÷ 2. If the x-coordinates are equal, the axis is horizontal, y = (y + y′) ÷ 2.
Drawing the image in a line
- From each vertex, draw a line perpendicular to the axis and extend it beyond the axis.
- Measure the distance from the vertex to the axis and mark the same distance on the other side.
- Join the image points in the same order.
Describe a reflection by stating the axis (for example “reflection in the line y = 1”), not just the word “reflection”.
(a) Find the image of (−6, 2) in the y-axis. (b) Q(−1, 3) is mapped to Q′(5, 3). State the axis.
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(a) (6, 2)
(b) Midpoint of −1 and 5 is (−1 + 5) ÷ 2 = 2, so the axis is x = 2.
(b) Midpoint of −1 and 5 is (−1 + 5) ÷ 2 = 2, so the axis is x = 2.
11.4Rotation
Rotation turns every point of a figure through a given angle and direction (clockwise or anticlockwise) about a fixed point called the centre of rotation.
To describe a rotation you must give three things: the centre, the angle and the direction.
- The image has the same shape, size and orientation as the object.
- The centre of rotation does not move.
- Each point of the image is the same distance from the centre as the object point.
| Rotation about the origin | (x, y) maps to | Example (3, 2) |
|---|---|---|
| 90° anticlockwise | (−y, x) | (−2, 3) |
| 180° (either direction) | (−x, −y) | (−3, −2) |
| 270° anticlockwise (= 90° clockwise) | (y, −x) | (2, −3) |
A rotation of 180° clockwise gives the same image as 180° anticlockwise. A rotation of 270° anticlockwise equals 90° clockwise.
Finding the image or the object
- Tracing paper: trace the object, hold the centre with the tip of a pencil, turn the paper through the angle.
- Protractor and ruler: join a point to the centre, measure the angle in the given direction and mark an equal length.
- Object from image: rotate the image the opposite way by the same angle.
- Centre: draw the perpendicular bisectors of two lines joining object and image points. They meet at the centre.
Example 8: rotation about a point that is not the origin
Rotate M(2, 5) through 90° anticlockwise about N(1, 2).Vector from N to M = (2 − 1, 5 − 2) = (1, 3). Rotating gives (−3, 1).
Image = N + (−3, 1) = (1 − 3, 2 + 1) = (−2, 3)
Example 9: object from the image
The image of K is K′(3, −1) under a rotation of 90° clockwise about (1, 1). Find K.Rotate K′ 90° anticlockwise about (1, 1): vector (2, −2) becomes (2, 2). K = (1 + 2, 1 + 2) = (3, 3).
When the centre is not the origin, first move the centre to the origin (subtract it), rotate, then add the centre back.
Find the image of (−4, 1) under a rotation of 180° about the origin. Then find the image of (3, 2) under 90° anticlockwise about the origin.
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180°: (−x, −y) = (4, −1).
90° anticlockwise: (−y, x) = (−2, 3).
90° anticlockwise: (−y, x) = (−2, 3).
11.5Translation, reflection and rotation as an isometry
Isometry is a transformation that keeps the distance between any two points of the object. The image is congruent to the object. Translation, reflection and rotation are all isometries.
| Transformation | Shape and size | Orientation | What is needed to describe it |
|---|---|---|---|
| Translation | Same | Same | Vector |
| Reflection | Same | Reversed | Axis of reflection |
| Rotation | Same | Same | Centre, angle, direction |
Because the image is congruent to the object under an isometry, corresponding lengths and angles are equal. A figure that is bigger or smaller (different size) or has a different shape is not an isometric image.
Example 10: identify the isometry
Triangle P has vertices (1, 1), (4, 1), (1, 3).Image with vertices (6, 1), (9, 1), (6, 3): translation 50.
Image with vertices (1, −1), (4, −1), (1, −3): reflection in the x-axis.
Image with vertices (−1, 1), (−1, 4), (−3, 1): rotation of 90° anticlockwise about the origin.
Example 11: using congruence
Triangle ABC is mapped to triangle PQR by an isometry. AB = 7 cm, ∠A = 52° and ∠B = 64°.PQ = AB = 7 cm. ∠R = ∠C = 180° − 52° − 64° = 64°.
One transformation after another
When a figure is transformed more than once, do the transformations in the order given, using the image of each step as the object of the next. Changing the order can give a different final image.
Example 12: two transformations in order
Point (3, 1) is translated by 12 and then reflected in the x-axis.After the translation: (3 + 1, 1 + 2) = (4, 3). After the reflection: (4, −3).
Reflecting first gives (3, −1), then translating gives (4, 1). This is different, so the order matters.
Name congruent figures with corresponding vertices in the same order, for example ABCD and SRQP.
A shape is moved so that every length is doubled. Is this an isometry? Why?
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No. The distance between two points changes, so the image is not congruent to the object.
11.6Rotational symmetry
A shape has rotational symmetry if it looks exactly the same after a rotation of less than 360° about a fixed point. The number of times it fits its original outline in one full turn is the order of rotational symmetry.
| Shape | Order of rotational symmetry | Smallest turn |
|---|---|---|
| Equilateral triangle | 3 | 120° |
| Square | 4 | 90° |
| Rectangle (not a square) | 2 | 180° |
| Regular hexagon | 6 | 60° |
For a regular polygon with n sides, the order is n and the smallest turn is 360° ÷ n. For these shapes the order equals the number of axes of symmetry.
A shape with order 1 has no rotational symmetry (only the full turn matches). Do not count a shape that only looks the same after 360° as having rotational symmetry.
State the order of rotational symmetry of (a) a regular pentagon (b) a regular octagon.
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(a) 5 (smallest turn 360° ÷ 5 = 72°)
(b) 8 (smallest turn 45°)
(b) 8 (smallest turn 45°)