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IGCSE / O Level Mathematics · Topic 7 Vectors and transformations · Chapter 32 · pp. 452–470

Transformations

A transformation moves or resizes a shape. The starting shape is the object and the new shape is the image. There are four kinds: reflection, rotation, translation and enlargement, and each needs particular details to be described fully.

Learning objectives

  1. Reflect a shape in a mirror line such as x = 2, y = 0, y = x or y = −x.
  2. Find the position and equation of the mirror line from an object and its image.
  3. Rotate a shape about a centre by 90° or 180° in a stated direction, and find the centre, angle and direction.
  4. Translate a shape by a column vector and find the vector between two shapes.
  5. Enlarge a shape from a centre by a positive, fractional or negative scale factor, and find the centre and scale factor.
  6. Describe a combination of two transformations.

Book exercises: 32.1 to 32.10 and Student assessments 1 to 3 (pp. 452–470). Enlargement with a negative scale factor is Extended content.

Four transformations at a glance

TransformationMovementTo describe it you give
ReflectionflipMirror line
RotationturnCentre, angle, direction
TranslationslideColumn vector
EnlargementresizeCentre, scale factor

💡 Reflection, rotation and translation keep the shape the same size (the image is congruent). Only enlargement changes the size.

Reflection

🔑 Key ideas

  • The object is “flipped” over a dashed mirror line.
  • Each point and its image are the same distance from the mirror line, measured at right angles.
  • The line joining a point to its image is perpendicular to the mirror line, and the mirror line cuts it in half.
  • Equations: x = k is a vertical line, y = k is horizontal, y = x and y = −x are the diagonals. The x-axis is y = 0 and the y-axis is x = 0.
-5-5-3-3-1-111335577ABCA'B'C'
Triangle ABC reflected in y = x (dashed). Each dashed link is cut in half at right angles by the mirror line, and the coordinates swap: (2, 1) goes to (1, 2).
Mirror linePoint (x, y) goes to
x-axis (y = 0)(x, −y)
y-axis (x = 0)(−x, y)
y = x(y, x)
y = −x(−y, −x)
x = k(2k − x, y)
y = k(x, 2k − y)

📘 Example 1 · Reflect P(5, 2) in the line x = 2

  1. x = 2 is a vertical line. P is 5 − 2 = 3 units to its right.
  2. The image is 3 units to the left of the line: 2 − 3 = −1.
  3. The height does not change, so y stays 2. Image: (−1, 2).
  4. Check with the table: (2k − x, y) = (4 − 5, 2) = (−1, 2). ✔

📘 Example 2 · A(1, 4) is reflected to A′(7, 4). Find the mirror line

  1. The mirror line cuts AA′ in half, so it passes through the midpoint.
  2. Midpoint: ((1 + 7) ÷ 2, (4 + 4) ÷ 2) = (4, 4).
  3. AA′ is horizontal, so the mirror line is perpendicular to it: vertical.
  4. A vertical line through x = 4 has equation x = 4.

📘 Example 3 · Reflect (1, 5) in the line y = x + 2

  1. y = x + 2 is a diagonal line through (0, 2) sloping up at 45°.
  2. Subtract the 2 so the line becomes y = x: the point (1, 5) becomes (1, 3).
  3. Swap the coordinates for the reflection in y = x: (3, 1).
  4. Add the 2 back: (3, 3).
  5. Check: the midpoint of (1, 5) and (3, 3) is (2, 4), and 4 = 2 + 2 so it lies on the mirror line. ✔

✏️ Try it

  1. Reflect (3, 2) in the line x = 5.
  2. Reflect (−4, 1) in the y-axis.
  3. Reflect (2, 5) in the line y = x.
Answers

1. 2k − x = 10 − 3 = 7, so (7, 2) · 2. (4, 1) · 3. swap to (5, 2)

Next: rotation →

Rotation

🔑 Key ideas

  • The object is “turned” about a fixed point, the centre of rotation.
  • Three details are needed: centre, angle, direction (clockwise or anti-clockwise).
  • Every point stays the same distance from the centre. A point and its image are joined to the centre by lines of equal length that meet at the angle of rotation.
  • Turning 90° clockwise is the same as 270° anti-clockwise. A 180° turn needs no direction.
-5-5-3-3-1-111335577ABCA'B'C'centre
Triangle ABC rotated 90° clockwise about the origin. A(1, 1) goes to A′(1, −1) and B(4, 1) goes to B′(1, −4). The dashed lines from the centre to B and B′ are equal in length and at right angles.
Rotation about the origin(x, y) goes to
90° clockwise(y, −x)
90° anti-clockwise(−y, x)
180°(−x, −y)

💡 For a centre that is not the origin, measure the point from the centre first: (dx, dy) = (x − cx, y − cy). Rotate (dx, dy) with the table, then add the centre back.

📘 Example 1 · Rotate B(4, 1) through 90° clockwise about (1, 1)

  1. Measure from the centre: (dx, dy) = (4 − 1, 1 − 1) = (3, 0).
  2. Rotate 90° clockwise: (dx, dy) becomes (dy, −dx) = (0, −3).
  3. Add the centre back: (1 + 0, 1 − 3) = (1, −2).
  4. Check: B is 3 units right of the centre; the image is 3 units below it, which is a clockwise quarter turn. ✔

📘 Example 2 · A(3, 1) maps to A′(1, −3) and B(1, 1) maps to B′(1, −1). Describe the rotation

  1. The centre is the same distance from a point and its image, so it lies on the perpendicular bisector of AA′ and of BB′.
  2. BB′: from (1, 1) to (1, −1), so the bisector is the horizontal line y = 0.
  3. AA′: midpoint (2, −1), direction (−2, −4), so the bisector is x + 2y = 0.
  4. Where y = 0 meets x + 2y = 0: x = 0. The centre is (0, 0).
  5. A is to the right of the centre and A′ is below it, so the turn is clockwise. OA is at about 18° above the x-axis and OA′ at about 72° below it, so the angle is 90°.
  6. Answer: rotation of 90° clockwise about (0, 0).
Join two points to their images Draw bisectors they cross at the centre Compare angles angle and direction
Finding the centre of rotation: draw the perpendicular bisector of each point-and-image line.

✏️ Try it

  1. Rotate (3, 1) through 90° clockwise about the origin.
  2. Rotate (2, −5) through 180° about the origin.
  3. Rotate (4, 0) through 90° anti-clockwise about (1, 1).
Answers

1. (y, −x) = (1, −3) · 2. (−2, 5) · 3. (dx, dy) = (3, −1) becomes (−dy, dx) = (1, 3); add (1, 1): (2, 4)

Next: translation →

Translation

🔑 Key ideas

  • A translation is a straight “slide”. There is no turning or flipping, so the image looks exactly like the object.
  • Every point moves the same way, described by a column vector (x / y): x units right (negative means left), then y units up (negative means down).
  • To find the vector, subtract: image − object for any one pair of matching points.
  • The reverse translation has the opposite signs: (3 / −2) is undone by (−3 / 2).
-5-5-3-3-1-111335577ABCA'B'C'
Triangle ABC slides by the vector (6 / −4): 6 right, 4 down. Every corner moves along a parallel dashed arrow.

📘 Example 1 · A(1, 2) maps to A′(4, −1). Find the vector

  1. Right/left: 4 − 1 = 3, so 3 to the right.
  2. Up/down: −1 − 2 = −3, so 3 down.
  3. Vector: (3 / −3).
  4. Check by adding: (1 + 3, 2 − 3) = (4, −1). ✔

📘 Example 2 · Translate the triangle with corners (3, 1), (5, 1), (3, 4) by (−2 / 5)

  1. Add −2 to every x-coordinate and 5 to every y-coordinate.
  2. (3, 1) goes to (1, 6).
  3. (5, 1) goes to (3, 6).
  4. (3, 4) goes to (1, 9).
  5. Image corners: (1, 6), (3, 6), (1, 9). The side lengths are unchanged. ✔

✏️ Try it

  1. Translate (2, 3) by (4 / −5).
  2. Find the vector that maps (1, 1) to (−3, 4).
  3. What vector undoes the translation (−2 / 5)?
Answers

1. (6, −2) · 2. (−3 − 1 / 4 − 1) = (−4 / 3) · 3. (2 / −5)

Next: enlargement →

Enlargement

🔑 Key ideas

  • The image is similar to the object (same shape, different size).
  • Two details are needed: the centre of enlargement and the scale factor k.
  • Scale factor = (a side on the image) ÷ (the matching side on the object), or (distance from centre to image point) ÷ (distance from centre to object point).
  • k > 1: image bigger. 0 < k < 1: image smaller (still called an enlargement).
  • To find the centre, join each point to its image and extend the lines: they meet at the centre.
-5-5-3-3-1-111335577ABCA'B'C'O
Triangle ABC enlarged by scale factor 2 from O. Each dashed ray passes through a point and its image, and the image is twice as far from O: A(1, 1) goes to A′(2, 2).

📘 Example 1 · Find the centre and scale factor: A(2, 1), B(4, 1), C(2, 2) map to A′(3, 2), B′(7, 2), C′(3, 4)

  1. Line through A and A′: (2, 1) and (3, 2) gives y = x − 1.
  2. Line through B and B′: (4, 1) and (7, 2) gives y = (x − 1) ÷ 3.
  3. They meet where x − 1 = (x − 1) ÷ 3, so x = 1 and y = 0. Centre (1, 0).
  4. Scale factor: A′B′ ÷ AB = 4 ÷ 2 = 2.
  5. Check the third pair: C(2, 2) is 1 right and 2 up from (1, 0); doubled it is 2 right, 4 up, giving (3, 4) = C′. ✔

Negative scale factors Extended

🔑 Key ideas

  • With a negative scale factor, a point and its image are on opposite sides of the centre.
  • The image is upside down (the same as an enlargement of |k| followed by a 180° turn about the centre).
  • Work out the size as usual, then put a “−” before the number.
-5-5-3-3-1-111335577ABCDA'B'C'D'O
Scale factor −2 from O. The square ABCD is in the top right; its image A′B′C′D′ is twice the size and on the other side of O, in the bottom left.

📘 Example 2 · Enlarge A(2, 3) by scale factor −2 about the centre (1, 1)

  1. Measure from the centre: (2 − 1, 3 − 1) = (1, 2).
  2. Multiply by −2: (−2, −4).
  3. Add the centre back: (1 − 2, 1 − 4) = (−1, −3).
  4. Check: A and A′ are on opposite sides of (1, 1), and A′ is twice as far from the centre as A. ✔

📘 Example 3 · Enlarge (6, −3) by scale factor 1/3 about the origin

  1. The centre is the origin, so just multiply both coordinates by 1/3.
  2. 6 × 1/3 = 2 and −3 × 1/3 = −1.
  3. Image: (2, −1). It is smaller and nearer the centre.

✏️ Try it

  1. Enlarge (2, 3) by scale factor 3, centre the origin.
  2. Enlarge (3, 2) by scale factor 2, centre (1, 1).
  3. Enlarge (−2, 4) by scale factor −1/2, centre the origin.
Answers

1. (6, 9) · 2. (2, 1) from the centre, doubled (4, 2), plus (1, 1): (5, 3) · 3. (1, −2)

Next: combining transformations →

Combinations of transformations

🔑 Key ideas

  • An object can go through two or more transformations one after the other.
  • Do them in the order given. The image of the first becomes the object for the second.
  • Label the stages: ABC, then A′B′C′, then A″B″C″.
  • To describe a combination, describe each step in full.
-8-8-6-6-4-4-2-2224466881010ABCA'B'C'A"B"C"centre
Triangle ABC (small) is enlarged by scale factor 3 from (0, 7) to make A′B′C′, then reflected in the dashed line x = 1 to make A″B″C″.

📘 Example · A(1, 6), B(2, 6), C(1, 8) is enlarged by scale factor 3 from (0, 7), then reflected in x = 1

  1. Enlarge A: from the centre it is (1, −1); times 3 is (3, −3); add (0, 7): A′(3, 4).
  2. Enlarge B: (2, −1) times 3 is (6, −3); add (0, 7): B′(6, 4).
  3. Enlarge C: (1, 1) times 3 is (3, 3); add (0, 7): C′(3, 10).
  4. Reflect in x = 1 using (2 − x, y): A′ goes to A″(−1, 4), B′ goes to B″(−4, 4), C′ goes to C″(−1, 10).
  5. Check: A′ is 2 right of the line and A″ is 2 left of it. ✔
  6. Answer: A″(−1, 4), B″(−4, 4), C″(−1, 10).

🧪 Try it: transformation grid

Pale shape: object. Darker shape: image. Dashed lines join each point to its image.

Working

    ✏️ Try it

    1. Reflect (3, 1) in the x-axis, then translate by (2 / 3). Where does it end up?
    2. Rotate (2, 1) through 180° about the origin, then reflect in the y-axis.
    3. Which of the four transformations change the size of a shape?
    Answers

    1. (3, −1), then (5, 2) · 2. (−2, −1), then (2, −1) · 3. Only enlargement. Reflection, rotation and translation keep the size.

    Next: summary and quiz →

    Chapter summary

    • Reflection: give the mirror line. Points and images are equal distances from it, at right angles.
    • Rotation: give the centre, the angle and the direction.
    • Translation: give the column vector (right / up); vector = image − object.
    • Enlargement: give the centre and scale factor. Scale factor = image side ÷ object side. Find the centre by extending the lines through matching points.
    • Negative scale factor: the image is on the opposite side of the centre.
    • Reflection, rotation and translation keep the size; only enlargement changes it.
    • In a combination, do each step in order and label A, A′, A″.

    📄 Practice questions

    The book's Exercises 32.1 to 32.10 and Student assessments 1 to 3 are in the printable assignment for this chapter. Most use diagrams from the book, so keep the book open beside the sheet.

    Mini-quiz

    1. What is the image of (3, 2) after reflection in the y-axis?
    2. The point (2, 5) is reflected in the line x = 4. What is the image?
    3. A point goes from (1, 4) to (7, 4) in a reflection. What is the mirror line?
    4. Rotate (2, 1) through 90° clockwise about the origin. Where does it go?
    5. What is the image of (3, −2) after a rotation of 180° about the origin?
    6. Which three things describe a rotation fully?
    7. Translate (6, 1) by the vector (−4 over 5). What is the image?
    8. A shape is enlarged by scale factor 3 with centre (0, 0). Where does (2, −1) go?
    9. A side of length 6 cm becomes 15 cm in an enlargement. What is the scale factor?
    10. An enlargement has scale factor 1/2. What is true of the image?
    11. Enlarge (1, 2) by scale factor −2 with centre (0, 0). What is the image?
    12. The point (1, 3) is reflected in the line y = x. What is the image?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 32 (Transformations). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.