UNIT 12.6 LOCUS: A UNIT 12.6 LOCUS: A SET OF POINTSSET OF POINTS

Objectives Students are able to identify the locus of a set

of points that are: at a given distance d from a given point O at a given distance d from a given straight line equidistant from two given points equidistant from two given intersecting straight

lines locus of a set of points that satisfy the above

conditions using a compass, ruler and protractor a triangle given any three sides/angles using a

ruler, compass and protractor

Objectives Students are able to identify the locus

of a set of points that are >,<, ≥, ≤ a given distance d from a given

point O >,<, ≥, ≤ a given distance d from a given

straight line nearer to point A than point B nearer to line A than line B

Equidistant from two given points

A B

((

(

(

The locus is a perpendicular bisector of the line AB

At a given distance, d, from a given straight line

A B

The locus is a pair of lines parallel to the given line, AB at a distance d cm from AB

d

d

Equidistant from two given intersecting lines

( ((The locus is the angle bisector of the angle between the two intersecting lines

At a given distance, d, from a given point

AX

d The locus is a circle with center A, and radius d cm.

To be at right angle to a given line, AB

A B

The locus is a circle with center AB as the diameter of the circle

Example 1 Describe the locus of a point P, which moves in a plane so

that it is always 4cm from a fixed point O in the plane.

OX

4 cm The locus is a circle with

center O, and radius

4cm.

Example 2 Describe the locus of a point Q, which moves in a

plane, so that it is always 5 cm from a given straight line, l.

lThe locus is a pair of lines parallel to the given

line, l, at a distance 5 cm from it.

5 cm

5 cm

Example 3 Two points A and B are 7.5cm apart. Draw the locus of a point

P, equidistant from A and B.

A 7.5cm B

((

(

(

The locus is a

perpendicular bisector of the line AB

Example 4

Draw two intersecting lines l and m. Draw the locus of a point P which moves such that it is equidistance from l and m.

( ((The locus is the angle bisector of the angle between the two intersecting lines

l

m

Example 5 Construct an angle XYZ equal to 60. Draw the locus of a

point P, which moves such that it is equidistant from XY and YZ.

( ((The locus is the angle bisector of the angle between the two intersecting lines Z

(60

Y

X

Example 6 Construct the triangle ABC such that AB = 6cm,

BC = 7cm and CA = 8cm. Draw the locus of P such that P is equidistant from A and C.

A 6cm B

C

8cm

7cm

( (

((Locus of P

Example 7 Construct a triangle PQR in which QR = 8cm, angle RQP = 70

and segment RP = 9cm. Construct the locus which represents the points equidistant from PQ and QR.

R 8cm Q

P

9cm

( ((

Locus

(70

Example 8

Constructing 60 angleStep 1: Construct Arc 1

Step 2: Construct Arc 2

Step 3: Draw line from intersection of two arc

Example 9 Construction of circumcircle

((

((

((

(( Step 1: Draw perpendicular bisector of 1 side of triangle

Step 2: Draw perpendicular bisector of 2nd side of triangle

Step 3: Intersection of bisector will be the center of circle

Example 10:

Construction of Inscribed Circle

(((

( ((Step 1: Draw angle bisector on 1st angle of triangle

Step 2: Draw angle bisector of 2nd angle of triangle

Step 3: Intersection of angle bisector will be

the center of circle

Independent Practice-1

A long stick leans vertically against a wall. The stick then slides in such a way that its upper end describes a vertical straight line down the wall, while the lower end crosses the floor in a straight line at right angles to the wall. Construct a number of positions of the mid point of the stick and draw the locus.

Intersection of Loci

If two or more loci intersect at a point P, then P satisfies the conditions of the both loci simultaneously.

Example:

A B

(

(

(

(

6cm•The circle is 6cm from point A.

•The perpendicular bisector is at equidistant from point A and B.

X

YThe point X and Y are both at : i) 6 cm from A ii) Equidistant from point A and B

Do it Yourself!

Question 1

a) Construct and label triangle XYZ in which XY=10cm, YZ=7.5cm and angle XYZ = 60. Measure and write down the length of XZ.

b) On your diagram, construct the locus of a point(i) 6cm from point Y

(ii) equidistant from X and Z.

c) The point P, inside the triangle XYZ is 6cm from Y and equidistant from point X and Y.

(i) Label clearly, on your diagram, point P.

(ii) Measure and write down the length of PX.

X Y

Z

Do it Yourself!

Question 5A factory occupies a quadrilateral site ABCD in which

AB=110m, BAD=65, AD=90m, ADC=110 and DC=60m.

(a) Using a scale of 1cm to represent 10m, construct a plan of the quadrilateral ABCD. Measure ABC.

Two fuel storage tanks, T1 and T2 are located 30m from C and 15m from BD respectively.(b) On the same diagram, draw the locus which represents

all the points inside the quadrilateral which arei) 30m from C ii) 15m from D

(c) Mark clearly on your diagram, the positions of thetanks T1 and T2.

(d) By measurement, find the distance between T1 and T2.

Do it Yourself! (Continue)

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