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Class 9 Maths — Why constructions work

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

Name: ________________________

  1. 1.Two sides of a triangle are 6 and 9 cm. What is the smallest whole number the third side could be, in centimetres?

  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.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?

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

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

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

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

    • 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.
    • Join the two crossing points with a straight line.

  6. 6.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?

  7. 7.Why must the compass opening stay the same when you swing arcs from each end of a segment?

    • a) So the finished line comes out the same length as the segment
    • b) So the finished line comes out horizontal
    • c) So each arc reaches both ends of the segment
    • d) So each crossing point is equally far from both ends

  8. 8.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
    • An angle of 60°
    • The perpendicular bisector of that segment
    • The bisector of that angle
    • An angle of 45°

Answer key — Class 9 Maths — Why constructions work

  1. 1. 4
  2. 2. 15
  3. 3. 14
  4. 4. 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
  5. 5. 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.
  6. 6. 60
  7. 7. d) So each crossing point is equally far from both ends
  8. 8. 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°