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## PowerPoint Slideshow about 'Locus' - adamdaniel

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Locus

Locust

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

path

Burp!

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 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.

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

3 cm

A

B

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?

The locusof 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

Perpendicular bisector

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 the blue pts.

Place your compass

at where the lines [walls] meet.

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:

C

C

(a)

(b)

3cm

3cm

2.5cm

A

O

B

A

O

B

D

D

The loci of a point 3cm from CD

The locus of a point 2.5cm from O

=> a circle of radius 2.5cm

with centre O

=> 2 straight lines // to CD at a

distance of 3cm from CD.

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 OB & OD

The locus of a point equidistant from C & O

=> the angle bisector of angle BOD

=> the perpendicular bisector of OC

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.

(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.

11.4cm

(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.

B

A

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

Y

X

(a)(iii)

(a) (i) YZ = 9cm

(a)(ii)

(a)(iv)

75o

45o

Y

X

(a)(iii)

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.

(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.

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

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

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

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

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

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

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

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

B

A

O

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 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.

C

locus of X

6cm

B

A

8cm

½x6xh >= 3

h >=1

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

LOCI CONSTRUCTION - Loci Involving Areas

Q5. (b) On your diagram, draw the locus of pts within

the triangle which are:

(i) 9cm from A,

Q5. (a) Draw ABC in which base AB = 12cm, ABC=50o

& BC = 7cm. Measure & write down the size of

ACB.

(b)(i)

(ii) 5.5cm from B,

(b)(ii)

(iii) 2.5cm from

AB,

C

(b)(iii)

7cm

50o

B

A

12cm

(a) ACB = 95o

½x12xh = 15

h =15/6

=2.5

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)

C

(b)(iii)

7cm

50o

12cm

B

A

possible position of P

(a) ACB = 95o

½x12xh >= 15

h >=15/6

>=2.5

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)

C

(b)(iii)

7cm

Region

of Q

50o

B

12cm

A

possible position of P

(a) ACB = 95o

Place your compass at Q & R.

LOCI CONSTRUCTION - Loci Involving Areas

Q6. Construct PQR in which PQ = 9.5cm, QPR=100o

& PR = 7.2cm.

(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)

100o

(b) Measure & write down

the radius of the circle.

Q

P

Radius = 6.5 cm

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

Q

P

Radius = 6.5 cm

A

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

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