Loci and Construction

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

Objectives

• Students are able to construct

– locus of a set of points that satify 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

A

X

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.

O

X

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.

l The 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 PQR =

70 and PR = 9cm. Construct the locus which represents

the points equidistant from PQ and QR.

R 8cm Q

P

9cm

Locus

70

Example 8

• Constructing 60 angle

Step 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

Question

• 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 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 Y The point X and Y are both at :

i) 6 cm from A ii) Equidistant from point A and B