Year 9 Maths The Thai Guy

Lesson 13.2 · 25 min

Describing Transformations

Giving every detail needed to describe a reflection, rotation, translation or enlargement — and finding mirror lines and centres of rotation.

In this lesson you will learn to…

  • Describe a single transformation fully
  • Find the equation of a mirror line
  • Find the centre and angle of a rotation
  • Write a translation as a column vector

Why this matters

“Turn it round a bit” is not an instruction anyone can follow exactly. Robot arms, animation software and CNC machines need precise descriptions: rotate 90° clockwise about (3,0)(3, 0). The same precision earns the marks in an exam.

Which transformation is it?

ClueTransformation
Same size, same way up, just movedtranslation
Same size, flipped over (mirror image)reflection
Same size, turnedrotation
Different sizeenlargement

Special mirror lines

LineEffect on (x,y)(x, y)
the xx-axis (y=0y = 0)(x,−y)(x, -y)
the yy-axis (x=0x = 0)(−x,y)(-x, y)
y=xy = x(y,x)(y, x)
y=−xy = -x(−y,−x)(-y, -x)

Rotation directions. A rotation of 90∘90^\circ clockwise is the same as 270∘270^\circ anticlockwise, and 180∘180^\circ needs no direction at all.

Key rules and formulas

  • What a full description needs

    Reflection — the mirror line (as an equation). Rotation — the centre, the angle and the direction. Translation — the column vector. Enlargement — the centre and the scale factor.

  • Mirror lines

    The mirror line is the perpendicular bisector of the line joining a point to its image. and reflect in ; and reflect in .

  • Column vectors

    The top number is the movement in (right is positive), the bottom number the movement in (up is positive).

  • Centre of rotation

    Construct the perpendicular bisectors of AA′ and BB′. They cross at the centre. Then measure the angle ACA′.

Worked examples

Worked example 1

Finding a mirror line

maps to under a reflection. Find the mirror line.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Identifying a transformation from coordinates

Triangle A(1, 2), B(4, 1), C(1, 5) becomes A′(−9, −2), B′(0, −5), C′(−9, 7). Describe the transformation.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

A translation

A(1, 2), B(2, 4), C(4, 3), D(5, 1) becomes A′(4, −2), B′(5, 0), C′(7, −1), D′(8, −3). Describe the transformation.

Try it yourself first, then reveal the solution one step at a time.

Practice: describing transformations

10 questions

Ten questions. Type a column vector as two numbers, top then bottom.
  1. Question 1 of 10Match the pairsFoundation
    Match each transformation to the information needed to describe it fully.

    Pick an answer, then choose the row it belongs to. You can also drag it.

    • Reflection
    • Rotation
    • Translation
    • Enlargement

  2. Question 2 of 10Complete the table
    Each point is reflected onto its image. Give the equation of the mirror line.
    ObjectImageMirror line
    (1, 5)(1, 3)
    (−3, 2)(7, 2)
    (−3, −1)(−3, 4)
    (6, 2)(2, 6)
  3. Question 3 of 10Fill the gaps
    A(1, 2), B(2, 4), C(4, 3), D(5, 1) becomes A′(4, −2), B′(5, 0), C′(7, −1), D′(8, −3). Give the translation vector.

    : top , bottom

  4. Question 4 of 10Multiple choice
    Triangle X(2, 3), Y(5, 4), Z(6, 1) becomes X′(−3, −2), Y′(−4, −5), Z′(−1, −6). What is the mirror line?
  5. Question 5 of 10Multiple choice
    A(2, 1), B(4, 0), C(6, 2) becomes A′(7, 1), B′(9, 0), C′(11, 2). Describe the transformation.
  6. Question 6 of 10Choose all that apply
    Select every complete description.

    Select every correct answer — there may be more than one.

  7. Question 7 of 10Fill the gapsChallenge
    A(1, 2), B(4, 1), C(1, 5) becomes A′(−9, −2), B′(0, −5), C′(−9, 7). It is an enlargement. Give its scale factor and centre.

    Scale factor , centre ( , )

  8. Question 8 of 10Type the answer
    Point is rotated about the origin. What is the -coordinate of its image?
  9. Question 9 of 10Put in order
    Put the steps for finding a centre of rotation in order.

    Put these in the right order. Drag them, or use the arrow buttons.

    1. Construct the perpendicular bisectors of AA′ and BB′
    2. Measure the angle ACA′ to find the angle of rotation
    3. Mark where the bisectors cross: this is the centre C
    4. Join A to its image A′ and B to its image B′

  10. Question 10 of 10True or false
    A rotation of clockwise is the same as a rotation of anticlockwise about the same centre.