Loci

1. 1 Draw [2 marks]

   Draw the locus of all the points that are 3 cm from a point \(P\).

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   Model answer

   The locus is a circle with centre \(P\) and radius 3 cm.

   Mark scheme

   • A circle — M1
   • With centre \(P\) and radius 3 cm — A1
2. 2 Shade [4 marks]

   The diagram shows a rectangular garden \(ABCD\), drawn to a scale of 1 cm to 1 m. A bush must be planted so that it is closer to \(AB\) than to \(AD\), and less than 6 m from \(A\). Show how to find the region where the bush could be planted. (4 marks)

   

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   Model answer

   Bisect the angle at \(A\), which makes a \(45^\circ\) line from \(A\). The region closer to \(AB\) is below this line. Draw an arc of radius 6 cm with centre \(A\). The region is inside the arc, below the bisector, and inside the rectangle.

   Mark scheme

   • Angle bisector at \(A\), with arcs — M1
   • Side nearer \(AB\) chosen — A1
   • Arc of radius 6 cm centred on \(A\) — M1
   • The correct region shaded, inside the rectangle — A1
3. 3 Draw [3 marks]

   \(A\) and \(B\) are two points 6 cm apart. Describe the locus of the points that are the same distance from \(A\) and from \(B\), and say how far from \(A\) it crosses the line \(AB\).

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   Model answer

   The locus is the perpendicular bisector of \(AB\). It crosses \(AB\) at its midpoint, which is 3 cm from \(A\).

   Mark scheme

   • The perpendicular bisector — M1
   • Of the line AB — A1
   • 3 cm from A — B1
4. 4 Draw [3 marks]

   \(PQ\) is a line of length 5 cm. Describe the locus of the points that are 2 cm from the line \(PQ\), and work out the length of the straight parts of it.

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   Model answer

   The locus is two straight lines, 2 cm from \(PQ\) on each side and parallel to it, joined by semicircles of radius 2 cm round \(P\) and \(Q\). Each straight part is 5 cm long, so the straight parts total \(2 \times 5 = 10\) cm.

   Mark scheme

   • Two parallel lines 2 cm from PQ — M1
   • Semicircles at both ends — A1
   • 10 cm — B1
5. 5 Shade [4 marks]

   \(P\) and \(Q\) are two points 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)

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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
6. 6 Work out [3 marks]

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

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   Model answer

   No. The points that are closer to \(B\) than \(A\) are more than 4 cm from \(A\), because the perpendicular bisector is 4 cm from \(A\). A point less than 3 cm from \(A\) is inside a circle that does not reach the bisector.

   Mark scheme

   • No — B1
   • The bisector is 4 cm from \(A\) — M1
   • The circle of radius 3 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.