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Year 9 Maths — Why constructions work

Explain what each compass step guarantees, and decide when a triangle can be constructed at all.

Name: ________________________

  1. 1.Put these steps in order to build the perpendicular bisector of a segment AB.

    • Join the two crossing points with a straight line.
    • Open the compasses to more than half of AB.
    • Keeping that same opening, put the point on B and draw two more arcs, crossing the first two.
    • With the point on A, draw an arc above AB and another below it.

  2. 2.An angle of 60° is bisected, and one of the halves is then bisected again. How big is the smallest angle now, in degrees?

  3. 3.You draw two arcs of the same radius, one from each end of a segment. What is true of the point where they cross?

    • a) It is one radius from the mid-point
    • b) It is the mid-point of the segment
    • c) It lies on the segment itself
    • d) It is the same distance from both ends

  4. 4.A triangle is to be built on a base of 8 cm, with the other two sides adding up to 12 cm. What would its perimeter be, in centimetres?

  5. 5.Sort each set of measurements by whether it pins down exactly one triangle.

    Groups: Fixes exactly one triangle · Does not fix one triangle

    • One side on its own
    • Three angles: 40°, 60° and 80°
    • Three angles: 60°, 60° and 60°
    • Three sides: 5, 6 and 7
    • Two sides and the angle between them
    • Two angles and the side between them

  6. 6.Match each set of compass moves to what it produces.

    • Equal arcs above and below, swung from each end of a segment
    • Equal arcs swung from two marks on the arms of an angle
    • An arc of the same radius stepped once round from a point
    • A perpendicular raised on a line, then bisected
    • The perpendicular bisector of that segment
    • An angle of 60°
    • An angle of 45°
    • The bisector of that angle

  7. 7.Swing an arc from a point, then from where it cuts the arm swing another arc with the compasses opened exactly as before, and join up. All three lengths come out equal, so the triangle is equilateral. What angle have you just built, in degrees?

  8. 8.A triangle can be built only if any two of its sides add up to more than the third. Two sides are 6 and 9 cm. What is the largest whole number the third side could be, in centimetres?

Answer key — Year 9 Maths — Why constructions work

  1. 1. 1. Open the compasses to more than half of AB. 2. With the point on A, draw an arc above AB and another below it. 3. Keeping that same opening, put the point on B and draw two more arcs, crossing the first two. 4. Join the two crossing points with a straight line.
  2. 2. 15
  3. 3. d) It is the same distance from both ends
  4. 4. 20
  5. 5. Three sides: 5, 6 and 7 → Fixes exactly one triangle; Two sides and the angle between them → Fixes exactly one triangle; Two angles and the side between them → Fixes exactly one triangle; Three angles: 60°, 60° and 60° → Does not fix one triangle; Three angles: 40°, 60° and 80° → Does not fix one triangle; One side on its own → Does not fix one triangle
  6. 6. Equal arcs above and below, swung from each end of a segment → The perpendicular bisector of that segment; Equal arcs swung from two marks on the arms of an angle → The bisector of that angle; An arc of the same radius stepped once round from a point → An angle of 60°; A perpendicular raised on a line, then bisected → An angle of 45°
  7. 7. 60
  8. 8. 14