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GCSE Maths · Higher · Chapter 7: Transformations, constructions and loci · Lesson 3

Bisectors, loci and plans

Grades 4–8About 50 minutesCalculator allowed35 marks
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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

    Translate by .

    [1 mark]
    Answer
  2. A2
    From chapter 7

    Two triangles have all three angles equal. Must they be congruent?

    [1 mark]
    Answer
  3. A3
    Key Stage 3

    Work out .

    [1 mark]
    Answer
  4. A4
    Key Stage 3

    Work out .

    [1 mark]
    Answer
B

Example, then your turn

Grades 4–6

Your teacher works through each example. Then try the one beside it. Leave every arc showing.

Example 1

Construct the perpendicular bisector of .

AB
Hint Open the compasses wider than half of .
  1. B1

    Construct the perpendicular bisector of .

    [2 marks]
    PQ
Example 2

Construct the bisector of angle .

BCA
Hint Start with an arc centred on .
  1. B2

    Construct the bisector of angle .

    [2 marks]
    QRP
C

Practice

Grades 5–7
  1. C1

    Use a ruler and compasses to construct the perpendicular from to the line .

    [2 marks]
    Pl
  2. C2

    Use a ruler and compasses to construct an angle of at .

    [3 marks]
    AB
  1. C3

    The solid is made from cubes. Sketch its plan and its front elevation.

    [2 marks]
    front
  2. C4

    The plan, front and side (from the right) elevations of a solid made of cubes are shown. How many cubes are there?

    [2 marks]
    PlanfrontFrontSide
    Answer
  3. C5

    The rectangle is drawn on a grid of squares. Draw the locus of the points outside it that are exactly one square width from it.

    [2 marks]
D

Exam-style questions

Grades 6–8
  1. D1

    A goat is tied by a rope m long to a ring. The ring can slide along a straight rail m long in the middle of a field. Ali says, "The goat can reach more than m of the field."

    8 m3 mgoatrail
    (a)

    Describe the shape of the region the goat can reach.

    [1 mark]
    (b)

    Show that Ali is wrong. You must show your working.

    [3 marks]
    (c)

    Work out the perimeter of the region, correct to significant figures.

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

    The plan shows a garden on a grid of m squares. A sprinkler at waters every point within m of . A tree is to be planted where it is watered, at least m from the hedge , and no further from than from .

    ABCDShedge
    (a)

    Shade the region where the tree can be planted.

    [4 marks]
    (b)

    Work out the greatest possible distance of the tree from the hedge . Give your answer correct to decimal place.

    [2 marks]
    Answer
    Total for D2: 6 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 ladder m long leans against a vertical wall. Its foot slides away from the wall until the ladder lies flat on the floor. Describe the locus of the midpoint of the ladder, and work out how far moves, correct to significant figures.

    [4 marks]
    MwallfloorO
    Not drawn accurately
    Hint 1 · What to try

    Draw the ladder in three or four positions and mark each time. What shape do the points make?

    Hint 2 · The first line

    In a right-angled triangle, the midpoint of the longest side is the same distance from all three corners.

    Hint 3 · The full method

    So m always. moves on a quarter circle of radius m, centre : its length is .

    Answer ; m
F

How did it go?

Colour one circle on each line.
Not yetNearlyYes
I can construct bisectors, perpendiculars and angles of and with a ruler and compasses.
I can draw a locus and shade a region that meets several conditions.
I can draw plans and elevations, and count the cubes in a solid from them.

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