locus locust. locus the path of an object that obeys a certain condition
TRANSCRIPT
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Locus
Locust
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Locus
The path of an object that obeys a certain condition.
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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
Burp!
path
Specific condition
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If cows run on 2 legs…………..
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A cow runs on a straight level road. How will the locus of the cow look like?
locus
Specific condition
path
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2 loci that you will encounter often are circles and straight lines
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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.
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5 cm C
The locus of the cow is a circle with centre P & radius 5 m.
P
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The goat moves such that it is always 3 m away from the bar.How will the locus of the goat look like?
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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].
A B
3 cm 3 cm
3 cm 3 cm
We will be using scaled drawing here too =]
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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.
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Please, help me 5B!!! What should I do to make sure I am always exactly the same distance from both Strippy and Moppy?
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S M
Place your compass
at S.
Place your compass
at M.
Perpendicular bisector
The locus of Ms Chia is a perpendicular bisector of the line which joins Strippy [Point S] to Moppy [Point M].
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Ms Chia’s safest route
Strippy
Moppy
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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.
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The locus of canyon is the angle bisector of angle created when the 2 walls [2 lines] meet.
Place your
compass at where the lines [walls] meet.
Place your
compass at the
blue pts.
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Exams Tips
• 1 point Locus
Locus
Locus
Locus• 1 line
• 2 points
• 2 lines
Circle
2 parallel lines
Perpendicular bisector
Angle bisector
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LOCI CONSTRUCTION - Loci in 2 dimensions
2 straight lines AB & CD intersect at right angles at point O. Draw & describe in each diagram:
The locus of a point 2.5cm from O
A B
C
D
A B
C
D
(a) (b)
O O
=> a circle of radius 2.5cm with centre O
The loci of a point 3cm from CD
=> 2 straight lines // to CD at a distance of 3cm from CD.
2.5cm
3cm 3cm
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LOCI CONSTRUCTION - Loci in 2 dimensions
Q5. 2 straight lines AB & CD intersect at right angles at point O. Draw & describe in each diagram:
The locus of a point equidistant from C & O
A B
C
D
A B
C
D
(c) (d)
O O
=> the perpendicular bisector of OC
The locus of a point equidistant from OB & OD => the angle bisector of angle BOD
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• 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!
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LOCI CONSTRUCTION - Intersection of Loci
Q1. (a) Using ruler & compasses, construct ABC in which AB = 8.8cm, BC = 7cm & AC = 5.6cm.
A B
C
(b) On the same diagram, draw (i) the locus of a point which is 6.4cm from A
(i)
(ii)the locus of a point equidistant from BA & BC.
(ii)
(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
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LOCI CONSTRUCTION - Intersection of Loci
Q2. Construct & label XYZ in which XY = 8cm, YZX = 60o & XYZ = 45o.
X Y
Z
45o75o
(a) On your diagram, (i) measure & write down the length of YZ,
(a) (i) YZ = 9cm
(ii)draw the locus of a pt which is equidistant from X & Z,
(a)(ii)
(iii)draw the locus of a pt which is equidistant from ZX & ZY,
(a)(iii)
(iv) draw the locus of a pt which is 3cm from XY & on the same side of XY as Z,
(a)(iv)
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LOCI CONSTRUCTION - Intersection of Loci
Q2. Construct & label XYZ in which XY = 8cm, YZX = 60o & XYZ = 45o.
X Y
Z
45o75o
(a) (i) YZ = 9cm
(a)(ii)
(a)(iii)
(a)(iv)
(b) On your diagram, (i) label pt P which is equidistant from pts X & Z and from the lines ZX & ZY.
P
(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
(iii) measure & write down the length of PQ.
(b) (iii) PQ = 1cm
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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
O2cm P
P
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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
O2cm P
P
OP <=2cm
O2cm
P
OP < 2cm
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LOCI CONSTRUCTION - Further Loci (with shading)
Q1. (c) If OP > 2cm, the locus of P is the set of points the circle.outside
O2cm
P
P
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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
O2cm
P
P
circumference
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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
X Y
P
Place your compass at X & Y.
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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.
includingsolid
X Y
P
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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.
excludingbroken
X Y
P
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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.
O B2cm
A
The shaded region represents the locus of P where XY is the perpendicular bisector of AB
Y
X
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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.
X
B
X
A 8cm
6cm
C
Hint: Both triangles have the same height & base.
locus of X
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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.
LOCI CONSTRUCTION - Loci Involving Areas
If area of PQX >= 3cm2, ½x6xh >= 3 h >=1
1cm
Region in which X must lie
QP
RS
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Q5. (a) Draw ABC in which base AB = 12cm, ABC=50o
& BC = 7cm. Measure & write down the size of ACB.
LOCI CONSTRUCTION - Loci Involving Areas
A B
C
50o
12cm
7cm
Q5. (b) On your diagram, draw the locus of pts within the triangle which are: (i) 9cm from A,
(a) ACB = 95o
(b)(i) (ii) 5.5cm from B,
(b)(ii)
(iii) 2.5cm from AB, (b)(iii)
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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.
LOCI CONSTRUCTION - Loci Involving Areas
A
C
50o
12cm
7cm
(a) ACB = 95o
(b)(i)(b)(ii)
(b)(iii)
B
If area of PAB = 15cm2,½x12xh = 15 h =15/6 =2.5
possible position of P
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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.
LOCI CONSTRUCTION - Loci Involving Areas
C
50o
7cm
(a) ACB = 95o
(b)(i)(b)(ii)
(b)(iii)
If area of QAB >= 15cm2,½x12xh >= 15 h >=15/6 >=2.5
possible position of P
A 12cm B
Region of Q
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Q6. Construct PQR in which PQ = 9.5cm, QPR=100o
& PR = 7.2cm.
LOCI CONSTRUCTION - Loci Involving Areas
P Q
100o
R
(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, (iii) the circle through P, Q & R
Radius = 6.5 cm
(a)(i)
(a)(ii)
(a)(iii)
Place your compass at P & R.
Place your compass at Q & R.
(b) Measure & write down the radius of the circle.
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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.
LOCI CONSTRUCTION - Loci Involving Areas
P Q
100o
R
Radius = 6.5 cm
(a)(i)
(a)(ii)
(a)(iii)
A
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• Special thanks to Ms Wong WL and Murderous Maths for use of certain pictures and slides.