[discia] GCSE Maths › Geometry & Measures › Loci and Construction

Loci and Construction

discia.org 10 questions · 22 marks
Name Date
  1. Either paper 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.

    CD
    (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 \(D\).

    Show answer
    (a) The perpendicular bisector of \(CD\)
    (b) Every point on the line is the same distance from \(C\) as it is from \(D\)
    Another like this
  2. Either paper 3 marks

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

    RSM14 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 \(TS\).

    Show answer
    (a) \(7\ \text{cm}\)
    (b) \(9\ \text{cm}\)
    Another like this
  3. Either paper 1 mark
    Towns A and B are shown on a scale diagram. B2 cmA Scale: 1 cm represents 50 miles. What does the shaded area represent? Tick one box.

    ☐ all the points at least 100 miles from A

    ☐ all the points within 100 miles of B

    ☐ all the points within 2 miles of A

    ☐ all the points at least 2 miles from B

    Show answer
    all the points within 100 miles of B
    Another like this
  4. Either paper 2 marks

    The point \(P\) is marked on the sheet.

    P

    Use a pair of compasses to construct the locus of points that are exactly 2 cm from \(P\).

    Show answer
    the whole circle with centre P and radius 2 cm
    P
    Another like this
  5. Either paper 3 marks

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

    XYZ30
    (a)

    State the name of the construction that has been drawn.

    (b)

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

    Find the size of angle \(XYW\).

    Show answer
    (a) The bisector of angle \(XYZ\)
    (b) \(15^\circ\)
    Another like this
  6. Either paper 3 marks

    Here is the locus of all the points that are exactly \(7\) cm from the point \(P\).

    P7 cm
    (a)

    Write down the mathematical name of this locus.

    (b)

    Find the total length of the locus.

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

    Show answer
    (a) A circle of radius \(7\) cm with its centre at \(P\)
    (b) \(14\pi\ \text{cm}\)
    Another like this
  7. Either paper 2 marks

    \(X\) is the point \((-3,\ -4)\) and \(Y\) is the point \((1,\ -4)\).

    Here are four points.

    \((0,\ 0) \qquad (-1,\ 0) \qquad (-2,\ -4) \qquad (0,\ -8)\)

    Exactly one of them is the same distance from \(X\) as it is from \(Y\).

    Write down that point.

    Show answer
    \((-1,\ 0)\)
    Another like this
  8. Either paper 2 marks

    The points \(P\) and \(Q\) are 7 cm apart.

    PQ

    Use the compass and the straightedge to construct the locus of points that are the same distance from \(P\) as from \(Q\).

    You must show your construction arcs.

    Show answer
    the perpendicular bisector of PQ: arcs of one radius from P and from Q that cross on both sides, and the line through the two crossings
    PQ
    Another like this
  9. Either paper 2 marks

    The diagram shows two edges of a field, \(BA\) and \(BC\), which meet at the corner \(B\).

    ABC

    A path across the field starts at \(B\) and is always the same distance from \(BA\) as from \(BC\).

    Use the compass and the straightedge to construct the path. Show your construction arcs.

    Show answer
    the bisector of the 100° angle ABC: an arc from B that cuts BA and BC, arcs of one radius from the two cuts that cross, and a line from B through the crossing
    ABC
    Another like this
  10. Either paper 2 marks

    \(P\) is a point on the line \(AB\), with \(AP = 6\) cm and \(PB = 3\) cm.

    PAB

    Use the compass and the straightedge to construct an angle of \(90^\circ\) at \(P\).

    You must show all your construction arcs.

    Show answer
    a line at right angles to AB through P, 6 cm from A: arcs from P that cut AB on each side of it, arcs of one radius from those two points that cross, and a line from P through the crossing
    PAB
    Another like this
discia.org Loci and Construction · sheet 1
Answers Loci and Construction · sheet 1
  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. all the points within 100 miles of B
  4. the whole circle with centre P and radius 2 cm
    P
  5. (a) The bisector of angle \(XYZ\)
    (b) \(15^\circ\)
  6. (a) A circle of radius \(7\) cm with its centre at \(P\)
    (b) \(14\pi\ \text{cm}\)
  7. \((-1,\ 0)\)
  8. the perpendicular bisector of PQ: arcs of one radius from P and from Q that cross on both sides, and the line through the two crossings
    PQ
  9. the bisector of the 100° angle ABC: an arc from B that cuts BA and BC, arcs of one radius from the two cuts that cross, and a line from B through the crossing
    ABC
  10. a line at right angles to AB through P, 6 cm from A: arcs from P that cut AB on each side of it, arcs of one radius from those two points that cross, and a line from P through the crossing
    PAB
discia.org Loci and Construction · sheet 1