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Exam questions · Maths · Vectors, Constructions and Loci

Loci

  • 6 exam questions
  • 19 marks
  • 9 quick checks
  1. 1 Draw [2 marks]

    Draw the locus of all the points that are 2 cm from a point \(T\). [2 marks]

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    Model answer

    The locus is a circle with centre \(T\) and radius 2 cm.

    Mark scheme

    • A circle — M1
    • With centre \(T\) and radius 2 cm — A1
  2. 2 Shade [4 marks]

    The diagram shows the line \(XY\), which is 6 cm long. Shade the region of points that are less than 2 cm from the line \(XY\) and closer to \(X\) than to \(Y\). [4 marks]

    Two points X and Y, 6 centimetres apart.
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    Model answer

    Draw the shape that is 2 cm from \(XY\): parallel lines 2 cm on each side with semicircular ends. Then construct the perpendicular bisector of \(XY\), which crosses at 3 cm. The region is the part of the shape on the \(X\) side of the bisector.

    Mark scheme

    • Parallel lines 2 cm from XY with semicircular ends — M1
    • Perpendicular bisector of XY with arcs — M1
    • The \(X\) side chosen — A1
    • The correct region shaded — A1
  3. 3 Draw [3 marks]

    A square garden has sides of 10 m. A path must be the same distance from two adjacent sides, \(AB\) and \(AD\), starting at the corner \(A\). Describe the locus of points inside the garden that are the same distance from \(AB\) and \(AD\). [3 marks]

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    Model answer

    The locus is the bisector of angle \(A\), a straight line from \(A\) at \(45^\circ\) to both sides, which is the diagonal \(AC\) of the square.

    Mark scheme

    • The bisector of the angle at A — M1
    • At \(45^\circ\) to the sides — A1
    • The diagonal AC — B1
  4. 4 Draw [3 marks]

    A radio mast transmits to a distance of 30 km. A scale drawing uses 1 cm for 10 km. Describe the locus of the points that are exactly 30 km from the mast, and say how to draw it. [3 marks]

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    Model answer

    The locus is a circle with the mast at the centre. On the scale drawing the radius is \(30 \div 10 = 3\) cm.

    Mark scheme

    • A circle — M1
    • Centre at the mast — A1
    • Radius 3 cm on the drawing — B1
  5. 5 Shade [4 marks]

    \(A\) and \(B\) are two points 8 cm apart. Show how to find the region of points that are less than 5 cm from \(A\) and closer to \(B\) than to \(A\). [4 marks]

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    Model answer

    Draw a circle of radius 5 cm around \(A\) and the perpendicular bisector of \(AB\), which is 4 cm from \(A\). The region is inside the circle and on the \(B\) side of the bisector, a small segment of the circle.

    Mark scheme

    • Circle of radius 5 cm centred on \(A\) — M1
    • Perpendicular bisector with arcs — M1
    • The \(B\) side chosen — A1
    • The segment shaded — A1
  6. 6 Work out [3 marks]

    \(P\) and \(Q\) are 12 cm apart. Is there a point that is closer to \(Q\) than to \(P\) and less than 5 cm from \(P\)? Explain your answer. [3 marks]

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    Model answer

    No. The perpendicular bisector is 6 cm from \(P\), and points closer to \(Q\) are on the far side of it. A point less than 5 cm from \(P\) is inside a circle that does not reach the bisector.

    Mark scheme

    • No — B1
    • The bisector is 6 cm from \(P\) — M1
    • The circle of radius 5 cm does not reach it — A1

Quick check

  1. 1

    What is the locus of points 3 cm from a point \(P\)?

    1. AA circle with centre \(P\) and radius 3 cm
    2. BA straight line 3 cm long
    3. CA circle with diameter 3 cm
    4. DA square with sides 3 cm
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    A: A circle with centre \(P\) and radius 3 cm

    All points the same distance from \(P\) form a circle.

  2. 2

    What is the locus of points that are the same distance from two points \(A\) and \(B\)?

    1. AA circle through \(A\) and \(B\)
    2. BThe line \(AB\)
    3. CThe angle bisector at \(A\)
    4. DThe perpendicular bisector of \(AB\)
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    D: The perpendicular bisector of \(AB\)

    Every point on the perpendicular bisector is equidistant from \(A\) and \(B\).

  3. 3

    What is the locus of points that are the same distance from two lines that meet at a point?

    1. AA circle
    2. BThe perpendicular bisector of the lines
    3. CThe bisector of the angle between the lines
    4. DA line parallel to both
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    C: The bisector of the angle between the lines

    The angle bisector is equidistant from both arms.

  4. 4

    On a drawing, which region is “less than 3 cm from \(P\)”?

    1. AOutside the circle of radius 3 cm around \(P\)
    2. BInside the circle of radius 3 cm around \(P\)
    3. COn the circle only
    4. DA straight line through \(P\)
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    B: Inside the circle of radius 3 cm around \(P\)

    Less than 3 cm means closer than the circle, so inside it.

  5. 5

    What is the locus of points 2 cm from a line segment?

    1. AParallel lines 2 cm on each side with semicircular ends
    2. BA circle of radius 2 cm
    3. CA single parallel line
    4. DA rectangle with no rounded ends
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    A: Parallel lines 2 cm on each side with semicircular ends

    Near the ends, the points 2 cm away form semicircles.

  6. 6

    \(A\) and \(B\) are two points. Which side of the perpendicular bisector is “closer to \(A\) than \(B\)”?

    1. AThe side containing \(B\)
    2. BBoth sides
    3. CNeither side
    4. DThe side containing \(A\)
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    D: The side containing \(A\)

    Points nearer to \(A\) are on \(A\)'s side of the bisector.

  7. 7

    On a scale drawing with 1 cm for 1 m, a tree must be within 4 m of a post. What do you draw?

    1. AA circle of radius 4 m around the post
    2. BA line 4 cm long
    3. CA circle of radius 4 cm around the post
    4. DA circle of radius 1 cm
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    C: A circle of radius 4 cm around the post

    4 m is 4 cm on the drawing.

  8. 8

    A rectangle \(ABCD\) has \(A\) at the bottom left and \(B\) to its right. Which line separates the points closer to \(AB\) from those closer to \(AD\)?

    1. AThe diagonal \(BD\)
    2. BThe bisector of angle \(A\)
    3. CThe perpendicular bisector of \(AB\)
    4. DThe line \(AC\)
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    B: The bisector of angle \(A\)

    The angle bisector at \(A\) is the same distance from \(AB\) and \(AD\).

  9. 9

    \(A\) and \(B\) are 8 cm apart. Is there a point closer to \(B\) than \(A\) that is less than 3 cm from \(A\)?

    1. ANo
    2. BYes, any point inside the circle
    3. CYes, on the line \(AB\)
    4. DOnly on the circle
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    A: No

    The bisector is 4 cm from \(A\), and a circle of radius 3 cm around \(A\) does not reach it.