Practice Maths

Transformations

Key Ideas

Key Terms

Transformation
A movement or change applied to a shape; the original shape is the pre-image and the result is the image.
Translation
A slide — every point moves the same distance in the same direction, described by a vector (x, y).
Reflection
A flip over a mirror line — each point maps to an equal distance on the opposite side of the line.
Rotation
A turn about a fixed centre of rotation — described by the centre, angle, and direction (clockwise or anticlockwise).
Pre-image
The original shape before a transformation is applied.
Image
The resulting shape after a transformation has been applied.
Vector
A directed quantity used to describe a translation, written as (x, y) indicating horizontal and vertical movement.
Mirror line
The line of reflection — each point in the pre-image maps to an equal distance on the opposite side.
Centre of rotation
The fixed point about which a shape is rotated.
Invariant point
A point that does not move under a transformation — any point on the mirror line is invariant under reflection.

Describing Transformations

Translation: “3 units right and 2 units up” or as a column vector &binom{3}{2}.
Reflection: “reflection in the x-axis” or “reflection in the line y = 1”.
Rotation: “rotation of 90° anticlockwise about the origin.”

Hot Tip When finding the image of a rotation, it helps to rotate each vertex separately. Use tracing paper or count squares on a grid. Always state the centre, angle AND direction for a rotation.

Worked Example

Question: Point A is at (2, 3). Find its image after: (a) translation by vector &binom{4}{−1}, (b) reflection in the x-axis, (c) rotation of 90° anticlockwise about the origin.

(a) Translation: Add the vector to the coordinates.
A′ = (2 + 4, 3 + (−1)) = (6, 2)

(b) Reflection in x-axis: The x-coordinate stays the same; the y-coordinate changes sign.
A′ = (2, −3)

(c) Rotation 90° anticlockwise about origin: Rule: (x, y) → (−y, x)
A′ = (−3, 2)

The Four Transformations

A transformation moves or changes a shape in a defined way. The original shape is called the pre-image, and the result after the transformation is called the image. There are four types of transformations you need to know: translation, reflection, rotation, and dilation.

Three of these — translation, reflection, and rotation — are called isometric transformations (or rigid motions) because they preserve the size and shape of the figure. The image is congruent to the pre-image. A dilation, by contrast, changes the size of the figure.

Translation (Slide)

A translation slides every point of a figure the same distance in the same direction. Nothing is rotated or flipped — the shape simply moves. A translation is described using a vector, written as (x, y), which tells you how many units to move horizontally and vertically.

Example: Translate triangle ABC by vector (3, −2). Every vertex moves 3 units right and 2 units down. If A = (1, 4), then A' = (1+3, 4−2) = (4, 2).

Under a translation, the image has the same orientation as the pre-image — no flipping or turning occurs.

Reflection (Flip)

A reflection flips a figure over a line of reflection. Every point in the pre-image is mapped to a point the same distance on the opposite side of the mirror line, perpendicular to it.

Common lines of reflection: the x-axis (y = 0), the y-axis (x = 0), the line y = x, or any other straight line. When reflecting over the x-axis, the y-coordinate changes sign: (x, y) → (x, −y). When reflecting over the y-axis: (x, y) → (−x, y).

The image is congruent to the pre-image but has the opposite orientation — like a left hand reflected in a mirror looks like a right hand.

Rotation (Turn)

A rotation turns a figure through a given angle about a fixed point called the centre of rotation. A rotation is described by: the centre of rotation, the angle of rotation (in degrees), and the direction (clockwise or anticlockwise).

Common rotations about the origin: 90° anticlockwise: (x, y) → (−y, x). 180°: (x, y) → (−x, −y). 90° clockwise: (x, y) → (y, −x).

Under a rotation, every point in the figure stays the same distance from the centre of rotation. The shape and size do not change — only the orientation.

Dilation (Enlargement and Reduction)

A dilation makes a figure larger or smaller by a scale factor k, relative to a fixed centre of dilation. If k > 1, the image is larger (enlargement). If 0 < k < 1, the image is smaller (reduction). If k = 1, the image is the same size.

Under a dilation, all lengths are multiplied by k, but all angles remain unchanged. The image is similar to the pre-image (same shape, different size) but not congruent (unless k = 1).

To find the image of a point under a dilation with centre O and scale factor k: draw a ray from O through the point, and place the image point at k times the distance from O.

Key tip: When describing a transformation precisely, always give all the necessary information. For a translation: state the vector. For a reflection: state the mirror line. For a rotation: state the centre, angle, and direction. For a dilation: state the centre and scale factor. Missing any of these details means the transformation is not fully described, and you will lose marks in an exam.

Mastery Practice

  1. Find the image of each point after the given translation. Fluency

    1. A(1, 2), translate by &binom{3}{4}
    2. B(5, −1), translate by &binom{−2}{3}
    3. C(−3, 4), translate by &binom{5}{−2}
    4. D(0, −6), translate by &binom{−4}{6}
    5. E(7, 7), translate by &binom{−7}{−7}
    6. F(−2, −5), translate by &binom{6}{8}
    7. G(4, 0), translate by &binom{0}{−3}
    8. H(−1, 3), translate by &binom{−3}{−5}

  2. Find the image of each point after the given reflection. Fluency

    1. A(3, 5), reflect in the x-axis.
    2. B(−2, 4), reflect in the y-axis.
    3. C(6, −3), reflect in the x-axis.
    4. D(−4, −1), reflect in the y-axis.
    5. E(2, 2), reflect in the line y = x. (Rule: swap x and y)
    6. F(5, −3), reflect in the line y = x.
    7. G(4, 1), reflect in the line x = 2.
    8. H(1, 5), reflect in the line y = 3.

  3. Find the image of each point after the given rotation about the origin. Fluency

    Rotation rules: 90° anticlockwise: (x,y)→(−y,x) • 90° clockwise: (x,y)→(y,−x) • 180°: (x,y)→(−x,−y)

    1. A(3, 1), 90° anticlockwise
    2. B(4, −2), 90° clockwise
    3. C(−1, 5), 180°
    4. D(2, 3), 90° clockwise
    5. E(−3, −4), 90° anticlockwise
    6. F(0, 6), 180°
    7. G(5, 0), 90° anticlockwise
    8. H(−2, 4), 90° clockwise

  4. Fully describe the single transformation that maps shape A to its image A′. Understanding

    1. A has vertices (1,1), (3,1), (2,3). A′ has vertices (4,3), (6,3), (5,5).
    2. A has vertices (1,2), (4,2), (4,5). A′ has vertices (1,−2), (4,−2), (4,−5).
    3. A has vertices (2,1), (5,1), (5,4). A′ has vertices (−2,1), (−5,1), (−5,4).
    4. A has vertices (0,2), (2,0), (0,−2). A′ has vertices (2,0), (0,−2), (−2,0). (Rotation about origin)
    5. A has vertices (1,0), (3,0), (3,3). A′ has vertices (0,−1), (0,−3), (−3,−3). (Rotation about origin)

  5. Apply two transformations in sequence. Give the final image coordinates. Understanding

    1. Point P(3, 2): first translate by &binom{2}{−3}, then reflect in the x-axis.
    2. Point Q(−1, 4): first reflect in the y-axis, then translate by &binom{3}{1}.
    3. Triangle with vertices A(1,0), B(3,0), C(3,2): translate by &binom{−4}{2}, then rotate 90° clockwise about the origin.
    4. Point R(2, 5): rotate 180° about the origin, then reflect in the x-axis. What single transformation is equivalent?

  6. Problem solving with transformations. Problem Solving

    1. A square tile with vertices at (0,0), (2,0), (2,2), (0,2) is to be used to tile a floor by translating.
      1. Write the translation vector to place the next tile immediately to the right.
      2. Write the translation vector to place the tile in the second row, first column.
      3. Write the coordinates of the tile in position (row 3, column 2).
    2. A company logo is a triangle with vertices A(1,1), B(4,1), C(4,5). The designer creates a mirror-image version.
      1. Find the vertices of the image after reflection in the y-axis.
      2. Find the vertices of the image after reflection in the line x = 5.
    3. A flag is described by a polygon. When reflected in a vertical mirror line and then translated 6 units right, it returns to its original position. What is the equation of the mirror line?
    4. Explain why a translation, reflection and rotation all produce congruent images. What property is preserved in each case?
  7. Triangle Transformations. A triangle undergoes a series of transformations.

    Triangle T has vertices A(2, 1), B(5, 1), C(5, 4). Apply each transformation and give the image vertices. Problem Solving

    1. Rotate T by 90° anticlockwise about the origin. Label the image T′ and write all three image vertices.
    2. Reflect T in the line y = x. Label the image T′′ and write all three image vertices.
    3. Translate T by vector &binom{−3}{2}, then reflect in the y-axis. Write the final image vertices.
    4. After which single transformation (from parts (a), (b) or (c)) does the triangle end up in the second quadrant (all x-values negative, all y-values positive)?
  8. Equivalent Transformations. Some combinations of transformations equal a single transformation.

    Find the single transformation equivalent to each combination. Problem Solving

    1. Reflect point P(3, 5) in the x-axis, then reflect in the x-axis again. What single transformation is this equivalent to?
    2. Rotate point Q(4, 2) by 90° anticlockwise about the origin, then by another 90° anticlockwise. What single transformation is equivalent?
    3. Translate point R(1, 3) by &binom{4}{−2}, then translate by &binom{−4}{2}. What single transformation is this?
    4. Reflect point S(2, −1) in the x-axis, then reflect in the y-axis. What single transformation is equivalent?
  9. Symmetry and Invariance. Some points stay in place under certain transformations.

    Answer each question about invariant points and symmetry. Problem Solving

    1. Which points are invariant (stay in the same place) under a reflection in the x-axis? Describe all such points as a set.
    2. Which point is always invariant under a rotation about the origin?
    3. Point P(a, b) is reflected in the line y = x. For which values of a and b does P map to itself?
    4. A square has vertices (0,0), (4,0), (4,4), (0,4). List all rotations about its centre (2, 2) that map the square onto itself. (Give angles and directions.)
  10. Design Challenge. Use transformation knowledge in a spatial reasoning context.

    Answer each design and reasoning question. Problem Solving

    1. A shape is translated by &binom{3}{4} and then by &binom{−5}{1}. Write a single translation vector that achieves the same result.
    2. A design is made by reflecting triangle ABC with vertices A(1, 0), B(3, 0), C(2, 3) over the y-axis to create A′B′C′. What type of quadrilateral is formed by the four points A, B, B′, A′? Give a reason.
    3. A tessellation pattern is created by translating a quadrilateral tile repeatedly. The tile has vertices at (0,0), (3,0), (4,2), (1,2). Write the translation vectors needed to place the tiles: (i) immediately to the right, (ii) immediately above, (iii) in the position that is 2 right and 1 up from the original.
    4. Explain why a reflection followed by a reflection in a different parallel line is equivalent to a translation. Use coordinates to illustrate with a specific example.
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