Mathematics questions, chapter 8

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Loci in Two Dimensions

41 marks. Use Practice to check each answer as you go, or Exam to answer everything first. In Section B, show your working; a ruler and a pair of compasses help you check the diagrams.

Section A: objective questions

12 marks

Choose the correct answer. Each question carries 1 mark.

Question 1

Which statement best describes a locus?

A fixed point on a plane
A straight line that joins two points
The trace formed by a set of points that satisfy certain conditions
The outline of a solid object

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A locus is the trace or path formed by all the points that satisfy certain conditions. It can be a straight line, an arc or a curve.

For the teacher: Remembering, low difficulty, standard 8.1.1

Textbook p. 200

Question 2

A durian falls straight down from a tree. What is the locus of a point on the durian?

A straight line
A circle
An arc
A curve that bends upwards

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The durian moves straight downwards, so any point on it traces a vertical straight line. A circle or an arc needs a turning motion about a fixed point.

For the teacher: Understanding, low difficulty, standard 8.1.1

Textbook p. 201

Question 3

Diagram 1 shows a semicircular board fixed to the pole MN along its diameter. The board is rotated 360° about MN. What three-dimensional shape is formed?

A semicircular board with its straight edge on the vertical pole MN, N at the top and M at the bottom — Diagram 1

Cylinder
Cone
Hemisphere
Sphere

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Rotating a semicircle 360° about its diameter sweeps out a whole sphere. A hemisphere is only half of this solid: it is formed by a quarter circle turned about one of its straight edges.

For the teacher: Understanding, low difficulty, standard 8.1.1

Textbook p. 203

Question 4

A point P always moves 4 cm from a fixed point O. What is the locus of P?

A circle of diameter 4 cm with centre O
A circle of radius 4 cm with centre O
A straight line 4 cm from O
A circle of radius 4 cm that passes through O

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The constant distance from O is the radius, so the locus is a circle with centre O and radius 4 cm. A circle that passes through O would have O on its edge, not at its centre.

For the teacher: Remembering, low difficulty, standard 8.2.1

Textbook p. 204

Question 5

What is the locus of a point that is equidistant from two fixed points M and N?

The perpendicular bisector of MN
A circle with MN as its diameter
A straight line parallel to MN
The straight line MN

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Points that are the same distance from M and from N lie on the perpendicular bisector of MN: a straight line through the midpoint of MN at 90° to it.

For the teacher: Remembering, low difficulty, standard 8.2.2

Textbook p. 206

Question 6

A straight road AB is very long. A point X always moves 2.5 cm from AB. What is the distance between the two lines that make the locus of X?

1.25 cm
2.5 cm
7.5 cm
5 cm

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The locus is a pair of lines parallel to AB, one on each side, each 2.5 cm from AB. The distance between them is 2.5 + 2.5 = 5 cm. 2.5 cm is the distance from AB to one line only.

For the teacher: Applying, low difficulty, standard 8.2.3

Textbook p. 208

Question 7

In Diagram 2, PQ and RS are parallel lines 10 cm apart. What is the locus of a point that is equidistant from PQ and RS?

Two horizontal parallel lines PQ above and RS below with a dashed vertical gap of 10 cm between them — Diagram 2

A line parallel to PQ, 10 cm from PQ
A line perpendicular to PQ through the midpoint of PQ
A line parallel to PQ, 5 cm from both PQ and RS
Two lines parallel to PQ, each 5 cm from PQ

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The locus is one straight line parallel to both lines and halfway between them: 10 ÷ 2 = 5 cm from each. A pair of lines would be the locus of a point 5 cm from one line.

For the teacher: Applying, low difficulty, standard 8.2.4

Textbook p. 210

Question 8

In Diagram 3, ∠QPR = 70°. Z is a point that is always the same distance from PQ and from PR. What angle does the locus of Z make with PQ?

Two lines PQ and PR meeting at P with an angle of 70 degrees between them — Diagram 3

70°
35°
140°
90°

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The locus is the angle bisector of ∠QPR. It cuts 70° into two equal parts, so it makes 70° ÷ 2 = 35° with PQ.

For the teacher: Understanding, low difficulty, standard 8.2.5

Textbook p. 211

Question 9

In Diagram 4, ABCD is a rectangle with AB = 8 cm and AD = 6 cm. X is a point that is always 4 cm from A. Y is a point that is equidistant from AB and DC. The loci of X and Y meet at Z inside the rectangle. What is the distance of Z from AD?

Rectangle ABCD 8 cm by 6 cm with an arc of radius 4 cm from A and a horizontal line halfway up; they meet at Z — Diagram 4

2.6 cm
3 cm
4 cm
5 cm

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The locus of Y is 3 cm above AB (half of 6 cm). Z is 4 cm from A, so its distance from AD is √(4² − 3²) = √7 ≈ 2.6 cm. 3 cm is the height of Z, 4 cm is the radius and 5 cm comes from adding the squares instead of subtracting.

For the teacher: Applying, medium difficulty, standard 8.2.6

Textbook p. 214

Question 10

A point is the same distance from the x-axis and the y-axis, and is less than 5 units from the origin O. Which point could it be?

(4, 4)
(−6, 6)
(3, 3)
(4, 2)

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The point must have equal coordinates (or opposite ones), so (4, 2) is out. Distance from O: (4, 4) is √32 ≈ 5.7 and (−6, 6) is √72 ≈ 8.5, both more than 5. (3, 3) is √18 ≈ 4.2, which is less than 5.

For the teacher: Analysing, medium difficulty, standard 8.2.6

Textbook p. 215

Question 11

In Diagram 5, Road 1 and Road 2 are parallel and 30 m apart. P is a pole on Road 1. A tree is to be planted equidistant from the two roads and 15 m from P. How many positions are possible?

Two parallel horizontal roads with a 30 m gap and a pole P on the lower road — Diagram 5

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Equidistant from the roads means on the middle line, 15 m from each road and so 15 m from P at the closest. The circle of radius 15 m about P only touches this line, at one point directly above P. One position.

For the teacher: Analysing, high difficulty, standard 8.2.7

Textbook p. 216

Question 12

Aiman wants to construct the locus of points equidistant from P and Q, where PQ = 8 cm. He sets his compasses to 3 cm and draws arcs from P and from Q. What happens?

The arcs cross, but the line is drawn too far from P
The arcs do not cross, so he cannot find points on the locus
The arcs cross at the midpoint of PQ
The locus is found but is only 3 cm long

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The compasses must be set to more than half of PQ, which is more than 4 cm. With 3 cm, an arc from P reaches only 3 cm and an arc from Q reaches only 3 cm, so they stop 2 cm short of meeting.

For the teacher: Evaluating, medium difficulty, standard 8.2.2

Textbook p. 207

Section B: structured questions

29 marks

Answer all questions. Give your reasons and show your working.

Question 13

Diagram 6 shows a line segment PQ of length 6 cm. X is a point that is always 2.5 cm from P. Y is a point that is always the same distance from P and from Q.

Line segment PQ, 6 cm long, with the end points P on the left and Q on the right — Diagram 6

[2 marks] (a)Describe the locus of X completely.

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A circle [1]1
with centre P and radius 2.5 cm [1]1

Tick each point you got.

[2 marks] (b)Describe the locus of Y completely.

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The perpendicular bisector of PQ [1]1
a straight line at 90° to PQ through its midpoint, 3 cm from P [1]1

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 8.2.2

Textbook p. 206

Question 14

Diagram 7 shows a line segment CD of length 8 cm. T is a point that is always 1.5 cm from the line segment CD.

Line segment CD, 8 cm long, with the end points C on the left and D on the right — Diagram 7

[2 marks] (a)Describe the locus of T completely.

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Two straight lines parallel to CD, each 8 cm long and 1.5 cm from CD [1]1
joined by a semicircle of radius 1.5 cm at C and at D [1]1

Tick each point you got.

[2 marks] (b)Calculate the total length of the locus of T. Use π = 3.14.

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Straight parts: 2 × 8 = 16 cm [1]1
The two semicircles make one circle: 2 × 3.14 × 1.5 = 9.42 cm, so the total is 16 + 9.42 = 25.42 cm [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 8.2.3

Textbook p. 213

Question 15

Diagram 8 shows a square ABCD with sides of 6 units on a square grid. X and Y are points that move inside the square. X is always the same distance from AB and from AD. Y is always 4 units from C.

Square ABCD on a 6 by 6 grid with A bottom-left, B bottom-right, C top-right and D top-left — Diagram 8

[1 mark] (a)Describe the locus of X.

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The bisector of ∠BAD, that is the diagonal AC [1]1

Tick each point you got.

[1 mark] (b)Describe the locus of Y.

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An arc of the circle with centre C and radius 4 units [1]1

Tick each point you got.

[2 marks] (c)The two loci meet at one point inside the square. Calculate the distance of this point from A, correct to 2 decimal places.

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AC = √(6² + 6²) = √72 = 8.49 units [1]1
The point is 4 units from C along CA, so its distance from A is 8.49 − 4 = 4.49 units [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 8.2.6

Textbook p. 214

Question 16

Diagram 9 is a map drawn to a scale of 1 cm to 200 m. Garden P and garden Q are 8 cm apart. The straight road AB is parallel to PQ and 2 cm from it. A clinic is to be built so that it is equidistant from P and Q and is 600 m from the road AB.

Map with garden P and garden Q 8 cm apart on a line and the road AB parallel to and below it — Diagram 9

[1 mark] (a)How many centimetres on the map is 600 m?

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600 ÷ 200 = 3 cm [1]1

Tick each point you got.

[2 marks] (b)Describe the two loci that the clinic must be on.

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The perpendicular bisector of PQ [1]1
A pair of lines parallel to AB, 3 cm from AB on each side [1]1

Tick each point you got.

[2 marks] (c)The council chooses the position that is closer to P. How far from garden P, in metres, is it?

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That position is 4 cm along from P and 1 cm from PQ: √(4² + 1²) = √17 = 4.12 cm [1]1
4.123 × 200 = 824.6, so about 825 m [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 8.2.7

Textbook p. 216

Question 17

Hana makes two statements about loci. Statement 1: "The locus of a point that is always 4 cm from a fixed point O is a circle with a diameter of 4 cm." Statement 2: "The locus of a point that is equidistant from two fixed points A and B is the straight line AB."

[2 marks] (a)Statement 1 is wrong. Explain her mistake and write the correct statement.

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4 cm is the distance from O to the point, so it is the radius, not the diameter [1]1
The locus is a circle with centre O and radius 4 cm (diameter 8 cm) [1]1

Tick each point you got.

[2 marks] (b)Statement 2 is wrong. Explain her mistake and write the correct statement.

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Only the midpoint of AB is the same distance from A and B; other points on the line AB are nearer to one end [1]1
The locus is the perpendicular bisector of AB [1]1

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 8.2.1

Textbook p. 204

Question 18

Diagram 10 shows three shops P, Q and R on a square grid with sides of 1 unit, where 1 unit represents 50 m. P is at (0, 0), Q is at (8, 0) and R is at (6, 6). The council wants to build a bus stop that is the same distance from all three shops.

Triangle PQR on a square grid with P bottom-left, Q at 8 units to the right of P and R 6 units up and 6 right of P — Diagram 10

[2 marks] (a)Using loci, explain how to find the position of the bus stop.

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Construct the perpendicular bisector of PQ and the perpendicular bisector of QR (or of PR) [1]1
The bus stop is at the point where the two bisectors cross [1]1

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[1 mark] (b)Explain why this point is the same distance from all three shops.

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It is on the bisector of PQ so it is as far from P as from Q, and on the bisector of QR so it is as far from Q as from R; so all three distances are equal [1]1

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[2 marks] (c)The bus stop is at (4, 2). Calculate its distance from P in metres, correct to the nearest metre.

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√(4² + 2²) = √20 = 4.472 units [1]1
4.472 × 50 = 223.6, so about 224 m [1]1

Tick each point you got.

For the teacher: Creating, high difficulty, standard 8.2.7

Textbook p. 216

Question 19

In Diagram 11, each shaded shape is rotated 360° about the pole MN. Name the three-dimensional shape formed in each case.

Three shapes on a pole MN: (a) a rectangle, (b) a right-angled triangle, (c) a semicircle, each with one side on the pole — Diagram 11

[1 mark] (a)Shape (a), a rectangle

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Right cylinder [1]1

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[1 mark] (b)Shape (b), a right-angled triangle

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Cone [1]1

Tick each point you got.

[1 mark] (c)Shape (c), a semicircle

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Sphere [1]1

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 8.1.1

Textbook p. 203

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