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Exam questions · Maths · Vectors, Constructions and Loci

Ruler-and-Compass Constructions

  • 6 exam questions
  • 16 marks
  • 9 quick checks
  1. 1 Construct [2 marks]

    Use ruler and compasses to construct an angle of \(60^\circ\) at the point \(A\) on a line. You must show all your construction lines.

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    Model answer

    Draw an arc with centre \(A\) that crosses the line at \(P\). With the same radius and centre \(P\), draw an arc that crosses the first arc at \(Q\). Join \(A\) to \(Q\). The angle \(PAQ\) is \(60^\circ\), because \(APQ\) is an equilateral triangle.

    Mark scheme

    • Arc from A and an arc of the same radius from the crossing point — M1
    • Line through \(A\) and the point where the arcs cross — A1
  2. 2 Construct [3 marks]

    The diagram shows a line \(AB\) of length 6 cm. (a) Use ruler and compasses to construct the perpendicular bisector of \(AB\). You must show all your construction lines. (2 marks) (b) Write down the distance from \(A\) to the point where the bisector crosses \(AB\). (1 mark)

    A line AB of length 6 centimetres.
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    Model answer

    (a) Open the compasses to more than 3 cm. Draw arcs of the same radius from \(A\) and from \(B\), crossing above and below the line. Join the two crossing points with a straight line. (b) The bisector crosses \(AB\) at its midpoint, 3 cm from \(A\).

    Mark scheme

    • (a) Arcs of equal radius from A and B, crossing above and below — M1
    • (a) A straight line through the crossing points, with the arcs left on — A1
    • (b) 3 cm — B1
  3. 3 Construct [3 marks]

    Use ruler and compasses to construct a triangle \(ABC\) with \(AB = 6\) cm, \(BC = 5\) cm and \(AC = 4\) cm. You must show all your construction lines.

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    Model answer

    Draw \(AB\) 6 cm long. With the compasses set to 4 cm and the point on \(A\), draw an arc. With the compasses set to 5 cm and the point on \(B\), draw an arc that crosses the first. Join the crossing point \(C\) to \(A\) and \(B\).

    Mark scheme

    • \(AB = 6\) cm drawn accurately — B1
    • Arc of radius 4 cm from \(A\) and arc of radius 5 cm from \(B\) — M1
    • Triangle completed, with \(C\) at the crossing — A1
  4. 4 Explain [2 marks]

    \(M\) is a point on the perpendicular bisector of \(AB\). Explain why \(MA = MB\).

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    Model answer

    The perpendicular bisector cuts \(AB\) in half at right angles. The two triangles formed by \(M\), \(A\), \(B\) and the midpoint have equal sides next to the right angle and a common side, so they are congruent and \(MA = MB\). More simply, every point on the perpendicular bisector is the same distance from \(A\) and \(B\).

    Mark scheme

    • States that the bisector cuts AB in half at a right angle — B1
    • Uses congruent triangles, or states that points on the bisector are equidistant — B1
  5. 5 Construct [3 marks]

    Use ruler and compasses to construct an angle of \(30^\circ\). You must show all your construction lines.

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    Model answer

    Construct an angle of \(60^\circ\) with two arcs of the same radius. Then bisect the \(60^\circ\) angle by drawing an arc on both arms, then matching arcs from those points, and a line through the crossing. This gives two angles of \(30^\circ\).

    Mark scheme

    • A construction of \(60^\circ\) — M1
    • A bisector construction with arcs of equal radius — M1
    • A \(30^\circ\) angle completed — A1
  6. 6 Construct [3 marks]

    Use ruler and compasses to construct an equilateral triangle with sides of 5 cm. You must show all your construction lines.

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    Model answer

    Draw a line \(AB\) of 5 cm. Set the compasses to 5 cm. Draw an arc from \(A\) and an arc from \(B\), crossing at \(C\). Join \(C\) to \(A\) and \(B\). All three sides are 5 cm.

    Mark scheme

    • A line of 5 cm drawn — B1
    • Two arcs of radius 5 cm from the ends — M1
    • Triangle completed — A1

Quick check

  1. 1

    What does the perpendicular bisector of a line do?

    1. ACuts it in half at any angle
    2. BIs parallel to it
    3. CCuts it in a ratio of \(2 : 1\)
    4. DCuts it in half at right angles
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    D: Cuts it in half at right angles

    Perpendicular means at right angles, and bisector means cuts in half.

  2. 2

    What must you leave on your drawing in a construction?

    1. ANothing, rub them out
    2. BOnly the final line
    3. CThe construction arcs
    4. DThe protractor marks
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    C: The construction arcs

    The arcs show the method, and earn the marks.

  3. 3

    Which instruments do you use for a construction?

    1. AA protractor only
    2. BA ruler and compasses
    3. CA calculator
    4. DA set square and a protractor
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    B: A ruler and compasses

    Constructions use a ruler and a pair of compasses.

  4. 4

    Which angle is constructed using two arcs of the same radius, as in an equilateral triangle?

    1. A\(60^\circ\)
    2. B\(45^\circ\)
    3. C\(90^\circ\)
    4. D\(30^\circ\)
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    A: \(60^\circ\)

    The triangle with three equal sides has three angles of \(60^\circ\).

  5. 5

    A \(60^\circ\) angle is bisected. What is the size of each part?

    1. A\(15^\circ\)
    2. B\(60^\circ\)
    3. C\(120^\circ\)
    4. D\(30^\circ\)
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    D: \(30^\circ\)

    \(60 \div 2 = 30\).

  6. 6

    Why must the compasses be opened to more than half the length of the line when constructing a perpendicular bisector?

    1. ASo that the line is longer
    2. BSo that the angle is \(90^\circ\)
    3. CSo that the arcs from both ends cross
    4. DIt does not matter
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    C: So that the arcs from both ends cross

    If the radius is too small the arcs do not meet.

  7. 7

    What is true of every point on an angle bisector?

    1. AIt is on the vertex
    2. BIt is the same distance from both arms
    3. CIt is perpendicular to both arms
    4. DIt is the same distance from the vertex
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    B: It is the same distance from both arms

    The bisector is equidistant from the two arms.

  8. 8

    A triangle has sides 6 cm, 5 cm and 4 cm. After drawing the 6 cm side \(AB\), how do you find \(C\) if \(AC = 4\) cm and \(BC = 5\) cm?

    1. AArc of radius 4 cm from \(A\) and arc of radius 5 cm from \(B\), where they cross
    2. BArc of radius 5 cm from \(A\) and arc of radius 4 cm from \(A\)
    3. CA line of length 9 cm
    4. DA \(90^\circ\) angle at \(A\)
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    A: Arc of radius 4 cm from \(A\) and arc of radius 5 cm from \(B\), where they cross

    The third corner is the point 4 cm from \(A\) and 5 cm from \(B\).

  9. 9

    What is the shortest distance from a point to a line?

    1. AThe distance to the nearest end
    2. BThe distance along a \(45^\circ\) line
    3. CThe distance to the midpoint
    4. DThe perpendicular distance
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    D: The perpendicular distance

    The shortest path to a line meets it at a right angle.