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Loci and Construction worksheet

10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

10 questions 28 marks Sheet 1
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  1. 2 marks

    \(CD\) is a straight line \(9\) cm long.

    The diagram shows a construction made on it with a pair of compasses and a straight edge. The construction arcs have been left in.

    A line segment with compass arcs swung from each end, crossing above and below, and the line drawn through the crossingsCD
    (a)

    Write down the name of the construction that has been drawn.

    (b)

    Write down what is true about the distance from any point on the drawn line to \(C\) and to \(D\).

  2. 3 marks

    \(RS\) is a straight line \(14\) cm long. The perpendicular bisector of \(RS\) has been constructed, and it crosses \(RS\) at \(M\).

    A line segment with compass arcs from each end and the perpendicular bisector drawn through their crossings, meeting the segment at MRSM14 cm
    (a)

    Write down the length of \(RM\).

    (b)

    The point \(T\) lies on the bisector, and \(TR = 9\) cm.

    Write down the length of \(TS\).

  3. 3 marks

    Here is angle \(DEF\), which is \(40^\circ\), with a construction drawn on it. The construction arcs have been left in.

    An angle with compass arcs cutting both arms and crossing between them, and the bisector drawn from the vertex through that crossingDEF40
    (a)

    State the name of the construction that has been drawn.

    (b)

    The bisector meets the arc between the arms at \(G\).

    Find the size of angle \(DEG\).

  4. 3 marks

    Here is the locus of all the points that are exactly \(9\) cm from the point \(C\).

    A point with the circle of all points a fixed distance from it drawn around itC9 cm
    (a)

    Write down the mathematical name of this locus.

    (b)

    Find the total length of the locus.

    Give your answer in terms of \(\pi\).

  5. 2 marks

    \(A\) is the point \((1,\ 0)\) and \(B\) is the point \((9,\ 0)\).

    Here are four points.

    \((8,\ 3) \qquad (5,\ 3) \qquad (2,\ 0) \qquad (8,\ -3)\)

    Exactly one of them is the same distance from \(A\) as it is from \(B\).

    Write down that point.

  6. 4 marks

    \(XY\) is a straight line \(5\) cm long.

    Here is the locus of all the points that are exactly \(2\) cm from the line \(XY\).

    A line segment surrounded by the locus of points a fixed distance from it, two parallel lines closed by a semicircle at each endXY2 cm
    (a)

    The locus is made from two straight parts and two curved parts.

    Write down the mathematical name of each curved part.

    (b)

    Find the perimeter of the locus.

    Give your answer in terms of \(\pi\).

  7. 3 marks

    The diagram is a scale drawing of a field containing a lighthouse, marked \(M\).

    The scale of the drawing is 1 cm represents \(60\) metres.

    The circle is the locus of points a fixed distance from the lighthouse.

    A point marked M with a circle drawn around it showing the locus of points a fixed distance awayM7 cm
    (a)

    Calculate the real distance, in metres, represented by the circle.

    (b)

    A second circle is drawn to show every point exactly \(300\) metres from the lighthouse.

    Write down the radius, in centimetres, of that circle on the drawing.

  8. 2 marks

    \(CD\) is a straight line \(6\) cm long.

    The diagram shows a construction made on it with a pair of compasses and a straight edge. The construction arcs have been left in.

    A line segment with compass arcs swung from each end, crossing above and below, and the line drawn through the crossingsCD
    (a)

    State the name of the construction that has been drawn.

    (b)

    Write down what is true about the distance from any point on the drawn line to \(C\) and to \(D\).

  9. 3 marks

    \(RS\) is a straight line \(6\) cm long. The perpendicular bisector of \(RS\) has been constructed, and it crosses \(RS\) at \(M\).

    A line segment with compass arcs from each end and the perpendicular bisector drawn through their crossings, meeting the segment at MRSM6 cm
    (a)

    State the length of \(RM\).

    (b)

    The point \(T\) lies on the bisector, and \(TR = 13\) cm.

    Write down the length of \(TS\).

  10. 3 marks

    The diagram shows angle \(XYZ\), which is \(30^\circ\), with a construction drawn on it. The construction arcs have been left in.

    An angle with compass arcs cutting both arms and crossing between them, and the bisector drawn from the vertex through that crossingXYZ30
    (a)

    State the name of the construction that has been drawn.

    (b)

    The bisector meets the arc between the arms at \(W\).

    Calculate the size of angle \(XYW\).

Answers

  1. (a) The perpendicular bisector of \(CD\)
    (b) Every point on the line is the same distance from \(C\) as it is from \(D\)
  2. (a) \(7\ \text{cm}\)
    (b) \(9\ \text{cm}\)
  3. (a) The bisector of angle \(DEF\)
    (b) \(20^\circ\)
  4. (a) A circle of radius \(9\) cm with its centre at \(C\)
    (b) \(18\pi\ \text{cm}\)
  5. \((5,\ 3)\)
  6. (a) A semicircle of radius \(2\) cm
    (b) \(10 + 4\pi\ \text{cm}\)
  7. (a) \(420\ \text{m}\)
    (b) \(5\ \text{cm}\)
  8. (a) The perpendicular bisector of \(CD\)
    (b) Every point on the line is the same distance from \(C\) as it is from \(D\)
  9. (a) \(3\ \text{cm}\)
    (b) \(13\ \text{cm}\)
  10. (a) The bisector of angle \(XYZ\)
    (b) \(15^\circ\)

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