1. Locus
Locust

2. Locus
The path of an object that obeys a certain condition.

3. Specific condition
A cow, grazing in a field, moves so that it is always a distance of 5m from the pole that it is tied to. How will the locus of the cow look like?
locus
path
Burp!

4. If cows run on 2 legs…………..

5. Specific condition
A cow runs on a straight level road. How will the locus of the cow look like?
path
locus

6. 2 loci that you will encounter often are circles and straight lines

7. Alamak! How to draw 5 m on paper?
A cow, grazing in a field, moves so that it is always a distance of 5m from the pole [P] that it is tied to. How will the locus of the cow [C] look like? P
Alamak! How to draw 5 m on paper?
Perform scale drawing! Let’s use 1 cm to represent 1 m.

8. The locus of the cow is a circle with centre P & radius 5 m.
5 cm C P

9. The goat moves such that it is always 3 m away from the bar.
How will the locus of the goat look like?

10. We will be using scaled drawing here too =]
The loci of the goat are 2 straight lines // to the bar [Line AB] at a distance of 3 m from the bar [Line AB].
3 cm
3 cm A B
3 cm
3 cm
We will be using scaled drawing here too =]

11. The very lovely Ms Chia is dashing off to meet her hunky fiance, but
as she was about to cut across the field, she spots Strippy on one
side and Moppy on the other. They are both looking hopefully in her
direction. She knows that whoever she passes closer to will
immediately assume that he’s invited to send her home. This is a huge
headache for Ms Chia.

12. Please, help me 5B!!! What should I do to make sure I am always exactly the same distance from both Strippy and Moppy?

14. Perpendicular bisector
The locus of Ms Chia is a perpendicular bisector of the line
which joins Strippy [Point S] to Moppy [Point M].
Place your compass at S.
Place your compass at M. S M
Stick the point of compass onto S. Then open up to a bit more than halfway to M. Draw an arc.
Stick the point onto M. Draw a second arc to cut the first one in two places.
Use the ruler to draw a straight line that goes through the 2 places where the arc crosses.
If you pick any point on this line, you will find it’s exactly the same distance from both S and M – so that is Ms Chia’s safest route!
Perpendicular bisector

15. Ms Chia’s
safest route
Strippy
Moppy

16. Suppose you created a canyon that can bring you to
outer space. Your canyon is magnetic. You must find a path that goes exactly between the 2 walls – one false move and your canyon will be dragged over to the side and splattered, WITH YOU ON IT.

17. at where the lines [walls] meet.
The locus of canyon is the angle bisector of angle created when the 2 walls [2 lines] meet.
Place your compass
at the blue pts.
Place your compass
at where the lines [walls] meet.

19. Exams Tips 1 point Circle 1 line 2 parallel lines 2 points
Locus
Circle
1 line
Locus
2 parallel lines
2 points
Locus
Perpendicular bisector
2 lines
Locus
Angle bisector

20. LOCI CONSTRUCTION - Loci in 2 dimensions
2 straight lines AB & CD intersect at right angles at
point O. Draw & describe in each diagram:
(a) C C
(b)
3cm
3cm
2.5cm A O B A O B D D
The locus of a point 2.5cm from O
The loci of a point 3cm from CD
=> a circle of radius 2.5cm
with centre O
=> 2 straight lines // to CD at a
distance of 3cm from CD.

21. LOCI CONSTRUCTION - Loci in 2 dimensions
Q5. 2 straight lines AB & CD intersect at right angles at
point O. Draw & describe in each diagram:
(c) C C
(d) A O B A O B D D
The locus of a point equidistant from C & O
The locus of a point equidistant from OB & OD
=> the perpendicular bisector of OC
=> the angle bisector of angle BOD

22. Additional links are put up on Wiki site so please explore
Reflection questions on Wiki
Whose turn is it to post question? Please get it done!

23. LOCI CONSTRUCTION - Intersection of Loci
Q1. (a) Using ruler & compasses, construct ABC
in which AB = 8.8cm, BC = 7cm & AC = 5.6cm.
(b) On the same diagram, draw
(i) the locus of a point which
is 6.4cm from A
(i)
(ii) C
(ii)the locus of a point
equidistant from
BA & BC.
(c) Find the distance
between 2 pts which
are both 6.4cm from
A & equidistant from
BA & BC. Give your ans in
cm, correct to 1 dec place.
11.4cm A B

24. LOCI CONSTRUCTION - Intersection of Loci
Q2. Construct & label XYZ in which XY = 8cm,
YZX = 60o & XYZ = 45o.
(a) On your diagram,
(i) measure & write down the length of YZ,
(ii)draw the locus of a pt which is equidistant from X & Z,
(iii)draw the locus of a pt which is
equidistant from ZX & ZY,
(a) (i) YZ = 9cm Z
(iv) draw the locus of a pt
which is 3cm from XY
& on the same side of
XY as Z,
(a)(ii)
(a)(iv)
75o
45o X Y
(a)(iii)

25. LOCI CONSTRUCTION - Intersection of Loci
Q2. Construct & label XYZ in which XY = 8cm,
YZX = 60o & XYZ = 45o.
(b) On your diagram,
(i) label pt P which is equidistant
from pts X & Z and from
the lines ZX & ZY. X Y Z
45o
75o
(a) (i) YZ = 9cm
(a)(ii)
(a)(iii)
(a)(iv)
(b) (iii) PQ = 1cm
(ii) label the pt Q which is
on the same side of
XY as Z, is
equidistant from X &
Z, & is 3cm from the
line XY. Q P
(iii) measure & write down
the length of PQ.

26. LOCI CONSTRUCTION - Further Loci (with shading)
Q1. (a) The locus of a point P whose distance from a
fixed point O is OP<= 2cm, is represented by the
points inside & on the of the
circle with centre O & radius 2 cm.
circumference P
2cm P O

27. LOCI CONSTRUCTION - Further Loci (with shading)
Q1. (b) If OP < 2cm, the locus of P will not include the
points on the circumference & the circumference
will be represented by a line.
broken P P
2cm
2cm P O O
OP <=2cm
OP < 2cm

28. LOCI CONSTRUCTION - Further Loci (with shading)
Q1. (c) If OP > 2cm, the locus of P is the set of points
the circle.
outside P P
2cm O

29. LOCI CONSTRUCTION - Further Loci (with shading)
Q1. (d) If OP >= 2cm, the locus of P is the set of points
the circle including the points
on the
outside
circumference P
2cm P O

30. Place your compass at X & Y.
LOCI CONSTRUCTION - Further Loci (with shading)
Q2. (a) If X and Y are 2 fixed pts and if a pt P moves in
a plane such that PX=PY, then the locus of P is
the ______________ ________ of the line XY.
perpendicular
bisector P
Place your compass at X & Y. X Y

31. LOCI CONSTRUCTION - Further Loci (with shading)
Q2. (b) If P moves such that PX <= PY, the locus of P is
the set of points shown in the shaded region
_______ all the pts on the perpendicular
bisector, which is represented by a ______ line.
including
solid P X Y

32. LOCI CONSTRUCTION - Further Loci (with shading)
Q2. (c) If P moves such that PX < PY, the locus of P is
the set of points shown in the shaded region
_______ all the pts on the perpendicular
bisector, which is represented by a ______ line.
excluding
broken P X Y

33. LOCI CONSTRUCTION - Further Loci (with shading)
Q3. The figure below shows a circle, centre O. The
diameter AB is 4cm long. Indicate by shading, the
locus of P which moves such that OP>= 2 cm & PA < PB. X
2cm A B O
The shaded region represents the locus of P where
XY is the perpendicular bisector of AB Y

34. LOCI CONSTRUCTION - Loci Involving Areas
Introduction:
The figure below shows a triangle ABC of area 24cm2.
Draw the locus of pt X, on the same side of AB as C
such that area of XAB = area of ABC.
Hint: Both triangles have
the same height & base. X X C
locus of X
6cm A
8cm B

35. LOCI CONSTRUCTION - Loci Involving Areas
Q4. The figure shows a rectangle PQRS
of length 6 cm & width 4 cm.
A variable pt X moves
inside the rectangle
such that
XP <= 4cm, XP>= XQ
& the area of
PQX >= 3cm2.
Construct & shade
the region in which
X must lie. R S
Region in
which X
must lie
1cm P Q
If area of PQX >= 3cm2,
½x6xh >= 3
h >=1

36. LOCI CONSTRUCTION - Loci Involving Areas
Q5. (a) Draw ABC in which base AB = 12cm, ABC=50o
& BC = 7cm. Measure & write down the size of
ACB.
Q5. (b) On your diagram, draw the locus of pts within
the triangle which are:
(i) 9cm from A,
(b)(i)
(ii) 5.5cm from B,
(b)(ii)
(iii) 2.5cm from
AB, C
(b)(iii)
7cm
50o A
12cm B
(a) ACB = 95o

37. LOCI CONSTRUCTION - Loci Involving Areas
Q5. (c) Mark & label on your diagram a possible position
of a pt P within triangle ABC such that AP <=9cm,
BP <= 5.5cm & area of PAB = 15cm2.
(b)(i)
(b)(ii)
If area of PAB = 15cm2,
½x12xh = 15
h =15/6
=2.5 C
(b)(iii)
7cm
50o
12cm A B
possible position of P
(a) ACB = 95o

38. LOCI CONSTRUCTION - Loci Involving Areas
Q5. (d) A pt Q is such that AQ >= 9cm, BQ <= 5.5 cm &
area QAB >=15cm2. On your diagram, shade the
region in which Q must lie.
(b)(i)
(b)(ii)
If area of QAB >= 15cm2,
½x12xh >= 15
h >=15/6
>=2.5 C
(b)(iii)
7cm
Region
of Q
50o A
12cm B
possible position of P
(a) ACB = 95o

39. LOCI CONSTRUCTION - Loci Involving Areas
Q6. Construct PQR in which PQ = 9.5cm, QPR=100o
& PR = 7.2cm.
Place your compass at Q & R.
(a)(iii)
(a) On the same diagram, draw
(i) the locus of a pt
equidistant from P & R,
(ii) the locus of a pt
equidistant from Q & R, R
(a)(i)
(iii) the circle through P,
Q & R
(a)(ii)
Place your compass at P & R.
100o
(b) Measure & write down
the radius of the circle. P Q
Radius = 6.5 cm

40. LOCI CONSTRUCTION - Loci Involving Areas
Q6. (c) A is the point on the same side of QR such that
AQR is isosceles, with QA=RA & QAR =100o.
Mark the point A clearly on your diagram.
(a)(iii) R
(a)(i)
(a)(ii)
100o P Q
Radius = 6.5 cm A

41. Special thanks to Ms Wong WL and Murderous Maths for use of certain pictures and slides.