Locust Locus The path of an object that

Locust

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

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

If cows run on 2 legs…………. .

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

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

A cow, grazing in a field, moves so that it is always a distance of 5 m 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.

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

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

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 A B 3 cm We will be using scaled drawing here too =]

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.

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

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. S M Place your compass at M. Perpendicular bisector

Ms Chia’s safest route Strippy Moppy

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.

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.

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

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) 3 cm 2. 5 cm A O B A O D D The locus of a point 2. 5 cm from O => a circle of radius 2. 5 cm with centre O The loci of a point 3 cm from CD => 2 straight lines // to CD at a distance of 3 cm from CD. B

LOCI CONSTRUCTION - Loci in 2 dimensions Q 5. 2 straight lines AB & CD intersect at right angles at point O. Draw & describe in each diagram: (c) A C O D The locus of a point equidistant from C & O => the perpendicular bisector of OC (d) B A C O D The locus of a point equidistant from OB & OD => the angle bisector of angle BOD B

• 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!

LOCI CONSTRUCTION - Intersection of Loci Q 1. (a) Using ruler & compasses, construct ABC in which AB = 8. 8 cm, BC = 7 cm & AC = 5. 6 cm. (b) On the same diagram, draw (i) the locus of a point which is 6. 4 cm from A (ii) C (ii)the locus of a point equidistant from BA & BC. 11. 4 cm (c) Find the distance between 2 pts which A are both 6. 4 cm from A & equidistant from BA & BC. Give your ans in cm, correct to 1 dec place. B

LOCI CONSTRUCTION - Intersection of Loci Q 2. Construct & label XYZ in which XY = 8 cm, YZX = 60 o & XYZ = 45 o. (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 (a) (i) YZ = 9 cm Z equidistant from ZX & ZY, (iv) draw the locus of a pt which is 3 cm from XY & on the same side of XY as Z, (a)(ii) (a)(iv) 75 o X 45 o (a)(iii) Y

LOCI CONSTRUCTION - Intersection of Loci Q 2. Construct & label XYZ in which XY = 8 cm, YZX = 60 o & XYZ = 45 o. (b) On your diagram, (i) label pt P which is equidistant from pts X & Z and from Z (a) (i) YZ = 9 cm the lines ZX & ZY. (b) (iii) PQ = 1 cm (ii) label the pt Q which is (a)(ii) (a)(iv) on the same side of XY as Z, is Q equidistant from X & Z, & is 3 cm from the P line XY. (iii) measure & write down 75 o 45 o the length of PQ. X (a)(iii) Y

LOCI CONSTRUCTION - Further Loci (with shading) Q 1. (a) The locus of a point P whose distance from a fixed point O is OP<= 2 cm, is represented by the points inside & on the circumference of the circle with centre O & radius 2 cm. P O 2 cm P

LOCI CONSTRUCTION - Further Loci (with shading) Q 1. (b) If OP < 2 cm, the locus of P will not include the points on the circumference & the circumference broken will be represented by a line. P P O 2 cm OP <=2 cm P O 2 cm OP < 2 cm

LOCI CONSTRUCTION - Further Loci (with shading) Q 1. (c) If OP > 2 cm, the locus of P is the set of points outside the circle. P P O 2 cm

LOCI CONSTRUCTION - Further Loci (with shading) Q 1. (d) If OP >= 2 cm, the locus of P is the set of points outside the circle including the points on the circumference. P O 2 cm P

LOCI CONSTRUCTION - Further Loci (with shading) Q 2. (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

LOCI CONSTRUCTION - Further Loci (with shading) Q 2. (b) If P moves such that PX <= PY, the locus of P is the set of points shown in the shaded region _______ including all the pts on the perpendicular solid line. bisector, which is represented by a ______ P X Y

LOCI CONSTRUCTION - Further Loci (with shading) Q 2. (c) If P moves such that PX < PY, the locus of P is the set of points shown in the shaded region _______ excluding all the pts on the perpendicular broken line. bisector, which is represented by a ______ P X Y

LOCI CONSTRUCTION - Further Loci (with shading) Q 3. The figure below shows a circle, centre O. The diameter AB is 4 cm long. Indicate by shading, the locus of P which moves such that OP>= 2 cm & PA < PB. X A O 2 cm B The shaded region represents the locus of P where XY is the perpendicular bisector of AB Y

LOCI CONSTRUCTION - Loci Involving Areas Introduction: The figure below shows a triangle ABC of area 24 cm 2. 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 6 cm A 8 cm B

LOCI CONSTRUCTION - Loci Involving Areas Q 4. The figure shows a rectangle PQRS of length 6 cm & width 4 cm. A variable pt X moves S inside the rectangle such that XP <= 4 cm, XP>= XQ & the area of PQX >= 3 cm 2. Construct & shade the region in which X must lie. P If area of PQX >= 3 cm 2, ½x 6 xh >= 3 h >=1 R Region in which X must lie 1 cm Q

LOCI CONSTRUCTION - Loci Involving Areas Q 5. (b) On your ABC diagram, drawbase the AB locus of pts within (a) Draw in which = 12 cm, ABC=50 o the which are: & BCtriangle = 7 cm. Measure & write down the size of (i) 9 cm from ACB. A, (b)(i) (ii) 5. 5 cm from B, (iii) 2. 5 cm from AB, (b)(ii) C (b)(iii) 7 cm 50 o A 12 cm B (a) ACB = 95 o

LOCI CONSTRUCTION - Loci Involving Areas Q 5. (c) Mark & label on your diagram a possible position of a pt P within triangle ABC such that AP <=9 cm, BP <= 5. 5 cm & area of PAB = 15 cm 2. (b)(i) If area of PAB = 15 cm 2, ½x 12 xh = 15 h =15/6 =2. 5 (b)(ii) C (b)(iii) A 7 cm 12 cm possible position of P 50 o B (a) ACB = 95 o

LOCI CONSTRUCTION - Loci Involving Areas Q 5. (d) A pt Q is such that AQ >= 9 cm, BQ <= 5. 5 cm & area QAB >=15 cm 2. On your diagram, shade the region in which Q must lie. (b)(i) If area of QAB >= 15 cm 2, ½x 12 xh >= 15 h >=15/6 >=2. 5 (b)(ii) C (b)(iii) 7 cm Region of Q 50 o A (a) ACB = 95 o 12 cm possible position of P B

LOCI CONSTRUCTION - Loci Involving Areas Q 6. Construct PQR in which PQ = 9. 5 cm, QPR=100 o & PR = 7. 2 cm. (a)(iii) (a) On the same diagram, draw (i) the locus of a pt Place your equidistant from P & R, compass at Q & R. (ii) the locus of a pt equidistant from Q & R, R (a)(i) (iii) the circle through P, Q&R (a)(ii) 100 o Place your (b) Measure & write down compass atof the circle. the radius P & R. Radius = 6. 5 cm P Q

LOCI CONSTRUCTION - Loci Involving Areas Q 6. (c) A is the point on the same side of QR such that AQR is isosceles, with QA=RA & QAR =100 o. Mark the point A clearly on your diagram. (a)(iii) R (a)(i) (a)(ii) 100 o Q P Radius = 6. 5 cm A

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