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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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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!

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If cows run on 2 legs…………..

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

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

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**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.

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

5 cm C 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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**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 =]

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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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**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

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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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**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.

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**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

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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: (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.

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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: (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

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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. (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

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**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)

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**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.

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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 P 2cm P O

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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 P P 2cm 2cm P O O OP <=2cm 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 P P 2cm O

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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 circumference P 2cm P O

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**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

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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. including solid P X Y

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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. excluding broken P X Y

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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. X 2cm A B O The shaded region represents the locus of P where XY is the perpendicular bisector of AB Y

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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. Hint: Both triangles have the same height & base. X X C locus of X 6cm A 8cm B

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**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

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**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

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**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

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**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

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**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

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**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

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

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