Transcript LOCI CONSTRUCTION - Free engineering e books

Locus
Locust
Locus
The path of an object that obeys a
certain condition.
Specific 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?
path
Burp!
If cows run on 2 legs…………..
Specific condition
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?
P
Alamak! How to draw 5 m on paper?
Perform scale drawing! Let’s use 1 cm to represent 1 m.
The locus of the cow is a circle with centre P & radius 5 m.
P
5 cm
C
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 locus of Ms Chia is a perpendicular bisector of the line
which joins Strippy [Point S] to Moppy [Point M].
Place your
compass at
S.
S
M
Place your
compass at
M.
Perpendicular bisector
Ms Chia’s
safest route
Strippy
Moppy
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 where
the lines
[walls]
meet.
Place your
compass
at the
blue pts.
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:
(a)
C
C
(b)
3cm
3cm
2.5cm
A
O
B A
O
D
D
The locus of a point 2.5cm from O
=> a circle of radius 2.5cm
with centre O
The loci of a point 3cm from CD
=> 2 straight lines // to CD at a
distance of 3cm from CD.
B
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)
A
C
O
D
The locus of a point equidistant
from C & O => the perpendicular
bisector of OC
(d)
B A
C
O
D
The locus of a point equidistant
from OB & OD => the angle
bisector of angle BOD
B
• 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)
(i) the locus of a point which
is 6.4cm from A
(ii)
C
(ii)the locus of a point
equidistant from
BA & BC.
11.4cm
(c) Find the distance
between 2 pts which
A
are both 6.4cm from
A & equidistant from
BA & BC. Give your ans in
cm, correct to 1 dec place.
B
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
(a) (i) YZ = 9cm
Z
equidistant from ZX & ZY,
(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
X
45o
(a)(iii)
Y
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
Z
(a) (i) YZ = 9cm
the lines ZX & ZY.
(b) (iii) PQ = 1cm
(ii) label the pt Q which is
(a)(ii)
(a)(iv)
on the same side of
XY as Z, is
Q
equidistant from X &
Z, & is 3cm from the
P
line XY.
(iii) measure & write down
75o
45o
the length of PQ.
X
(a)(iii)
Y
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 circumference of the
circle with centre O & radius 2 cm.
P
O
2cm
P
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
broken
will be represented by a
line.
P
P
O
2cm
OP <=2cm
P
O
2cm
OP < 2cm
LOCI CONSTRUCTION - Further Loci (with shading)
Q1. (c) If OP > 2cm, the locus of P is the set of points
outside
the circle.
P
P
O
2cm
LOCI CONSTRUCTION - Further Loci (with shading)
Q1. (d) If OP >= 2cm, the locus of P is the set of points
outside
the circle including the points
on the circumference .
P
O
2cm
P
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
_______
including all the pts on the perpendicular
solid line.
bisector, which is represented by a ______
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
_______
excluding all the pts on the perpendicular
broken line.
bisector, which is represented by a ______
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
A
O
2cm
B
The shaded region represents the
locus of P where XY is the
perpendicular bisector of AB
Y
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.
X
X
C
locus of X
6cm
A
8cm
B
LOCI CONSTRUCTION - Loci Involving Areas
Q4. The figure shows a rectangle PQRS
of length 6 cm & width 4 cm.
A variable pt X moves
S
inside the rectangle
such that
XP <= 4cm, XP>= XQ
& the area of
PQX >= 3cm2.
Construct & shade
the region in which
X must lie.
P
If area of
PQX >= 3cm2,
½x6xh >= 3
h >=1
R
Region in
which X
must lie
1cm
Q
LOCI CONSTRUCTION - Loci Involving Areas
Q5. (b)
On your ABC
diagram,
drawbase
the AB
locus
of pts within
(a) Draw
in which
= 12cm,
ABC=50o
the
which are:
&
BCtriangle
= 7cm. Measure
& write down the size of
(i) 9cm from
ACB.A,
(b)(i)
(ii) 5.5cm from B,
(iii) 2.5cm from
AB,
(b)(ii)
C
(b)(iii)
7cm
50o
A
12cm
B
(a) ACB = 95o
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)
If area of PAB
= 15cm2,
½x12xh = 15
h =15/6
=2.5
(b)(ii)
C
(b)(iii)
A
7cm
12cm
possible
position of P
50o
B
(a) ACB = 95o
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)
If area of
QAB >= 15cm2,
½x12xh >= 15
h >=15/6
>=2.5
(b)(ii)
C
(b)(iii)
7cm
Region
of Q
50o
A
(a) ACB = 95o
12cm
possible
position of P
B
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
Place your
equidistant from P & R,
compass at
Q & R.
(ii) the locus of a pt
equidistant from Q & R, R
(a)(i)
(iii) the circle through P,
Q&R
(a)(ii)
100o
Place your
(b) Measure
& write down
compass
the
radiusatof the circle.
P & R.
Radius = 6.5 cm
P
Q
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.