Secondary 4 - Locus

Locus The path of an object that obeys a certain 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 Burp! path Specific condition

If cows run on 2 legs…………..

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

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 5m from the pole [P] that it is tied to. How will the locus of the cow [C] look like? Alamak! How to draw 5 m on paper? Perform scale drawing! Let’s use 1 cm to represent 1 m. P

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

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 ]. A B 3 cm 3 cm 3 cm 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 5B!!! What should I do to make sure I am always exactly the same distance from both Strippy and Moppy?

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 ].

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 Locus Locus Locus 1 line 2 points 2 lines Circle 2 parallel lines Perpendicular bisector 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: 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

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

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 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

LOCI CONSTRUCTION - Intersection of Loci Q2. Construct & label XYZ in which XY = 8cm, YZX = 60 o & XYZ = 45 o . X Y Z 45 o 75 o (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)

LOCI CONSTRUCTION - Intersection of Loci Q2. Construct & label XYZ in which XY = 8cm, YZX = 60 o & XYZ = 45 o . (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 X Y Z 45 o 75 o (a) (i) YZ = 9cm (a)(ii) (a)(iii) (a)(iv)

LOCI CONSTRUCTION - F urther 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 O 2cm P P

LOCI CONSTRUCTION - F urther 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 O 2cm P P OP <=2cm O 2cm P OP < 2cm

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

LOCI CONSTRUCTION - F urther 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 O 2cm P P circumference

LOCI CONSTRUCTION - F urther 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.

LOCI CONSTRUCTION - F urther 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 X Y P

LOCI CONSTRUCTION - F urther 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 X Y P

LOCI CONSTRUCTION - F urther 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 B 2cm A The shaded region represents the locus of P where XY is the perpendicular bisector of AB Y X

LOCI CONSTRUCTION - Loci Involving Areas Introduction: The figure below shows a triangle ABC of area 24cm 2 . Draw the locus of pt X, on the same side of AB as C such that area of XAB = area of ABC. B A 8cm 6cm C Hint: Both triangles have the same height & base. locus of X X X

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 >= 3cm 2 . Construct & shade the region in which X must lie. LOCI CONSTRUCTION - Loci Involving Areas 1cm Region in which X must lie Q P R S If area of PQX >= 3cm 2 , ½x6xh >= 3 h >=1

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

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 = 15cm 2 . LOCI CONSTRUCTION - Loci Involving Areas A C 50 o 12cm 7cm (a) ACB = 95 o (b)(i) (b)(ii) (b)(iii) B possible position of P If area of PAB = 15cm 2 , ½x12xh = 15 h =15/6 =2.5

Q5. (d) A pt Q is such that AQ >= 9cm, BQ <= 5.5 cm & area QAB >=15cm 2 . On your diagram, shade the region in which Q must lie. LOCI CONSTRUCTION - Loci Involving Areas C 50 o 7cm (a) ACB = 95 o (b)(i) (b)(ii) (b)(iii) possible position of P A 12cm B Region of Q If area of QAB >= 15cm 2 , ½x12xh >= 15 h >=15/6 >=2.5

Q6. Construct PQR in which PQ = 9.5cm, QPR=100 o & PR = 7.2cm. LOCI CONSTRUCTION - Loci Involving Areas P Q 100 o 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) (b) Measure & write down the radius of the circle. Place your compass at P & R. Place your compass at Q & R.

Q6. (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. LOCI CONSTRUCTION - Loci Involving Areas P Q 100 o R Radius = 6.5 cm (a)(i) (a)(ii) (a)(iii) A

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

Secondary 4 - Locus

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Secondary 4 - Locus