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GCSE Maths · Higher · Chapter 7: Transformations, constructions and loci · Lesson 2

Transformations and combining them

Grades 4–8About 50 minutesNo calculator35 marks
Hints online, answers free with an account:
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NameClassDate
A

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

  1. A1
    Needed today

    Find the midpoint of and .

    [1 mark]
    Answer
  2. A2
    Last lesson

    Two triangles have two sides and the angle between them equal. Which condition shows they are congruent?

    [1 mark]
    Answer
  3. A3
    From chapter 5

    £500 grows by a year for 2 years, compounded. Find the final amount.

    [1 mark]
    Answer
  4. A4
    Key Stage 3

    Work out .

    [1 mark]
    Answer
B

Example, then your turn

Grades 4–7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Rotate the triangle clockwise about .

xy−3−2−112345−11234OC
Hint Turn each line from to a corner a quarter turn.
  1. B1

    Rotate the triangle anticlockwise about . Give the new vertices.

    [2 marks]
    xy−4−3−2−11234−2−1123OC
    Answer
Example 2

Enlarge the triangle by scale factor , centre .

xy−112345671234OC
Hint Multiply each step from to a corner by .
  1. B2

    Enlarge the triangle by scale factor , centre . Give the new vertices.

    [2 marks]
    xy−2−1123456−11234OC
    Answer
C

Practice

Grades 5–7
  1. C1

    Shape has rotational symmetry of order . Work out the smallest turn that maps onto itself. Shape first fits onto itself after . Give its order.

    [2 marks]
    Answer order
  2. C2

    A translation maps onto . Write down its column vector. The same translation maps onto . Find .

    [2 marks]
    Answervector
  1. C3

    Reflect triangle in the line . Give the vertices of the image. Which vertex of is invariant? Why?

    [3 marks]
    xy−112345−112345OAy = x
    Answer ; invariant
  2. C4

    Enlarge triangle by scale factor , centre . Write down the coordinates of the vertices of the image.

    [3 marks]
    xy−2−1123456−3−2−1123OTC
    Answer
D

Exam-style questions

Grades 6–8
  1. D1

    Triangle is shown on the grid.

    xy−3−2−1123−3−2−1123OP
    (a)

    Rotate triangle clockwise about . Label the image . Write down the coordinates of the vertices of .

    [2 marks]
    Answer
    (b)

    Reflect triangle in the -axis. Label the image . Write down the coordinates of the vertices of .

    [2 marks]
    Answer
    (c)

    Describe fully the single transformation that maps triangle onto triangle .

    [2 marks]
    Answer
    Total for D1: 6 marks
  1. D2

    Sam says, "Triangle is an enlargement of triangle with scale factor ."

    Triangles and are shown on a grid of centimetre squares.

    xy−5−4−3−2−112345−11234OAB
    (a)

    Explain why Sam is wrong, and say which enlargement does have scale factor .

    [2 marks]
    (b)

    Describe fully the single transformation that maps triangle onto triangle .

    [3 marks]
    Answer
    (c)

    Triangle is enlarged by scale factor , centre , to give triangle . Work out the area of triangle .

    [2 marks]
    Answer
    Total for D2: 7 marks
E

Extension

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

  1. E1

    A rotation of clockwise maps the point onto . Find the coordinates of the centre of the rotation. Then find the image of under the same rotation.

    [4 marks]
    Hint 1 · What to try

    The centre is the same distance from as from , and the turn about it is a quarter turn.

    Hint 2 · The first line

    Call the centre . The step from the centre to is . A quarter turn clockwise makes a step into .

    Hint 3 · The full method

    So . That gives and .

    Answercentre image
F

How did it go?

Colour one circle on each line.
Not yetNearlyYes
I can rotate, reflect, translate and enlarge a shape, and describe each one fully.
I can enlarge by a fractional or a negative scale factor from any centre.
I can replace two transformations by one, and find invariant points.

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