loci and construction - verulam.s3.amazonaws.comverulam.s3.amazonaws.com/resources/ks4/maths/year 11...
TRANSCRIPT
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
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