This document defines a locus as a set of points that satisfy certain geometric conditions. It provides examples of loci that are: a given distance from a point or line, equidistant from two points or lines, perpendicular or parallel to a given line, or satisfy other angle or distance criteria. The objectives are to identify loci using a compass, ruler, and protractor. Several examples are worked out step-by-step to illustrate how to construct loci for points satisfying different conditions. Independent practice problems are provided for students to construct their own loci diagrams.

1. UNIT 12.6 LOCUS: AUNIT 12.6 LOCUS: A
SET OF POINTSSET OF POINTS

2. 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

3. 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

4. Equidistant from two given points
A B
(
(
(
(
The locus is a perpendicular
bisector of the line AB

5. 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

6. Equidistant from two given intersecting lines
( ((
The locus is the angle
bisector of the angle between
the two intersecting lines

7. At a given distance, d, from a given point
A
X
d The locus is a circle with
center A, and radius d cm.

8. 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

9. 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.

10. 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

11. 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

12. 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

13. 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 linesZ
( 60°
Y
X

14. 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

15. 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°

16. Example 8
 Constructing 60° angle
Step 1: Construct Arc 1
Step 2: Construct Arc 2
Step 3: Draw line from
intersection of two arc

17. 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

18. 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

19. 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.

21. 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

22. 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.

23. X Y
Z

24. Do it Yourself!
Question 5
A 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.

25. 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 are
i) 30m from C ii) 15m from D
(c) Mark clearly on your diagram, the positions of the
tanks T1 and T2.
(d) By measurement, find the distance between T1 and T2.
Do it Yourself! (Continue)

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