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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 4 cm from a point \(P\). [2 marks]

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    Model answer

    The locus is a circle with centre \(P\) and radius 4 cm.

    Mark scheme

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

    The diagram shows two points \(P\) and \(Q\) that are 6 cm apart. Shade the region of points that are closer to \(P\) than to \(Q\) and are less than 4 cm from \(Q\). [4 marks]

    Two points P and Q, 6 centimetres apart.
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    Model answer

    Construct the perpendicular bisector of \(PQ\), which is 3 cm from each point, and draw a circle of radius 4 cm around \(Q\). The region is inside the circle and on the \(P\) side of the bisector.

    Mark scheme

    • Perpendicular bisector of PQ with arcs — M1
    • The \(P\) side chosen — A1
    • Circle of radius 4 cm centred on \(Q\) — M1
    • The correct region shaded — A1
  3. 3 Draw [3 marks]

    A rectangle \(ABCD\) has \(AB = 8\) cm and \(AD = 5\) cm. Describe the locus of points inside the rectangle that are the same distance from \(AB\) and from \(AD\). [3 marks]

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    Model answer

    The locus is the bisector of angle \(A\). It is a straight line from \(A\) at \(45^\circ\) to both sides, up to where it meets \(DC\) at 5 cm along.

    Mark scheme

    • The bisector of the angle at A — M1
    • At \(45^\circ\) to \(AB\) and \(AD\) — A1
    • Stopping at the side DC, 5 cm along — B1
  4. 4 Draw [3 marks]

    A tree is at the point \(T\). A scale drawing uses 1 cm to 1 m. Describe the locus of all the points that are exactly 2 m from the tree, and say how it is drawn. [3 marks]

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    Model answer

    The locus is a circle with centre \(T\). On the drawing the radius is 2 cm.

    Mark scheme

    • A circle — M1
    • Centre \(T\) — A1
    • Radius 2 cm on the drawing — B1
  5. 5 Shade [4 marks]

    \(A\) and \(B\) are two points 10 cm apart. Show how to find the region of points that are less than 6 cm from \(A\) and less than 6 cm from \(B\). [4 marks]

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    Model answer

    Draw a circle of radius 6 cm around \(A\) and a circle of radius 6 cm around \(B\). The region is where the two circles overlap, which is a lens-shaped region centred on the perpendicular bisector of \(AB\), which is 5 cm from each.

    Mark scheme

    • Circle of radius 6 cm centred on \(A\) — M1
    • Circle of radius 6 cm centred on \(B\) — M1
    • The overlap shaded — A1
    • Symmetrical about the perpendicular bisector — B1
  6. 6 Work out [3 marks]

    \(P\) and \(Q\) are 10 cm apart. Is there a point that is closer to \(Q\) than to \(P\) and less than 4 cm from \(P\)? Explain your answer. [3 marks]

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    Model answer

    No. The perpendicular bisector is 5 cm from \(P\), and the points closer to \(Q\) are on the far side of it. A point less than 4 cm from \(P\) is inside a circle that does not reach the bisector.

    Mark scheme

    • No — B1
    • The bisector is 5 cm from \(P\) — M1
    • The circle of radius 4 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.