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Exam questions · Maths · Vectors, Constructions and Loci

Ruler-and-Compass Constructions

  • 6 exam questions
  • 18 marks
  • 9 quick checks
  1. 1 Construct [3 marks]

    Use ruler and compasses to construct a triangle \(ABC\) with \(AB = 7\) cm, \(BC = 6\) cm and \(AC = 5\) cm. You must show all your construction lines. [3 marks]

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    Model answer

    Draw \(AB\) 7 cm long. With the compasses set to 5 cm and the point on \(A\), draw an arc. With the compasses set to 6 cm and the point on \(B\), draw an arc that crosses the first. Join the crossing point \(C\) to \(A\) and \(B\).

    Mark scheme

    • \(AB = 7\) cm drawn accurately — B1
    • Arc of radius 5 cm from \(A\) and arc of radius 6 cm from \(B\) — M1
    • Triangle completed — A1
  2. 2 Construct [3 marks]

    The diagram shows a straight line \(l\) and a point \(P\) that is not on the line. (a) Use ruler and compasses to construct the perpendicular from \(P\) to the line \(l\). You must show all your construction lines. [2 marks] (b) The point \(P\) is 3 cm above the line. Write down the shortest distance from \(P\) to \(l\). [1 mark]

    A line l and a point P above it.
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    Model answer

    (a) With the point on \(P\), draw an arc that cuts \(l\) twice, at \(X\) and \(Y\). From \(X\) and \(Y\) draw arcs of equal radius that cross below the line. Join \(P\) to the crossing point. (b) The shortest distance is the perpendicular distance, 3 cm.

    Mark scheme

    • (a) An arc from P that cuts the line twice, and matching arcs that cross — M1
    • (a) A straight line from \(P\) through the crossing — A1
    • (b) 3 cm — B1
  3. 3 Construct [3 marks]

    Use ruler and compasses to construct the perpendicular bisector of a line \(XY\) of length 8 cm. You must show all your construction lines. [3 marks]

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    Model answer

    Open the compasses to more than 4 cm. Draw arcs of equal radius from \(X\) and \(Y\) that cross above and below the line. Join the crossing points with a straight line.

    Mark scheme

    • Line \(XY\) of 8 cm drawn — B1
    • Arcs of equal radius from both ends — M1
    • A straight line through the crossing points — A1
  4. 4 Explain [2 marks]

    Explain why the compasses must be set to more than half the length of the line when constructing a perpendicular bisector. [2 marks]

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    Model answer

    The arcs from the two ends only meet if the radius is more than half the length of the line. If the radius is half or less, the arcs touch at the midpoint or do not meet, so there are no two crossing points to join.

    Mark scheme

    • The arcs must cross — B1
    • If the radius is half or less the arcs do not cross twice — B1
  5. 5 Construct [3 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. [3 marks]

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    Model answer

    With the point of the compasses on \(A\), draw an arc that crosses the line at \(P\). With the same radius and the point on \(P\), draw an arc that crosses the first arc at \(Q\). Join \(A\) to \(Q\). Triangle \(APQ\) is equilateral, so the angle \(PAQ\) is \(60^\circ\).

    Mark scheme

    • An arc from A that crosses the line — M1
    • An arc of the same radius from the crossing point — M1
    • A line through \(A\) and the crossing point — A1
  6. 6 Construct [4 marks]

    Use ruler and compasses to construct an angle of \(30^\circ\) at the point \(A\) on a line. You must show all your construction lines. [4 marks]

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    Model answer

    Construct \(60^\circ\) at \(A\) with two arcs of the same radius. Then bisect the \(60^\circ\) angle: draw an arc from \(A\) that cuts both arms, draw matching arcs from those points that cross, and join \(A\) to the crossing.

    Mark scheme

    • A construction of \(60^\circ\) — M1
    • An arc on both arms of the \(60^\circ\) angle — M1
    • Matching arcs that cross — M1
    • A line from \(A\) through the crossing — 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.