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School
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4
3
2
1
Grade 8
/ Other
9 exercises (including 3 corrected)
a
The Pythagorean triplets
(2 exercices)
The square roots
(1 exercice)
The GPS system and triangulation
(3 exercices)
1.
The
Pythagor
e
an
triplets
E.1951
The
triplet
a
;
b
;
c
)
of
in
tegers
is
said
to
b
e
Pythagorean
if
:
a
2
+
b
2
=
c
2
Here,
the
study
of
sev
eral
w
a
ys
to
obtain
Pythagorean
triplets
:
1
The
Pythagorean
form
ula
:
a
Complete
the
table
b
elo
w
:
n
2
n
+1
2
n
2
+2
n
2
n
2
+2
n
+1
1
2
5
b
Sho
w,
using
the
calculator,
that
for
:
n
=
1
;
n
=
2
;
n
=
5
The
triplets
of
in
tegers
:
2
n
+1
;
n
2
+2
n
;
2
n
2
+2
n
+1
are
Pythagorean
triplets.
2
Plato’s
form
ula
:
a
Complete
the
table
b
elo
w
:
n
2
n
n
2
−
1
n
2
+1
1
2
6
b
Sho
w,
using
the
calculator,
that
for
:
n
=
1
;
n
=
2
;
n
=
6
the
triplets
of
in
tegers
2
n
;
n
2
−
1
;
n
2
+1
are
Pythagorean
triplets.
3
Eculides’
form
ula
:
a
Complete
the
table
b
elo
w
:
m
n
2
mn
m
2
−
n
2
m
2
+
n
2
2
1
3
2
5
2
b
Sho
w,
using
the
calculator,
that
the
three
in
te-
ger
triplets
2
mn
;
m
2
−
n
2
;
m
2
+
n
2
are
Pythagorean
triplets.
E.1345
1
Using
double
distributivit
y
,
sho
w
the
follo
wing
equalit
y
:
(2
n
2
+
2
n
)(2
n
2
+
2
n
)
=
4
n
4
+
8
n
3
+
4
n
2
2
Giv
e
the
expanded
and
reduced
form
of
the
follo
wing
ex-
pression
:
(2
n
+
1)
2
(use
double
distributivity
with
(2
n
+1)(2
n
+1)
)
3
Establish
the
follo
wing
equalit
y
:
(2
n
2
+
2
n
+
1)
2
=
4
n
4
+
8
n
3
+
8
n
2
+
4
n
+
1
4
Sho
w
that
the
triplet
2
n
+1
;
2
n
2
+2
n
;
2
n
2
+2
n
+1
is
a
Pythagorean
triplet
for
an
y
p
ositiv
e
in
teger
n
.
2.
The
squar
e
r
o
ots
E.1956
Definition
:
(pr
ovisional)
Let
a
b
e
a
p
ositiv
e
n
um
b
er.
W
e
call
the
square
ro
ot
of
the
n
um
b
er
a
the
p
ositiv
e
n
um
b
er
suc
h
that
its
square
is
equal
to
a
.
Let’s
start
b
y
sho
wing
that
there
is
only
one
n
um
b
er
that
c
hanges
the
diagram
b
elo
w
:
1
Supp
ose
there
are
t
w
o
p
ositiv
e
n
um
b
ers
a
and
b
suc
h
that
a
2
=
b
2
.
a
Expand
and
reduce
the
expression
:
(
a
+
b
)(
a
−
b
)
.
b
What
can
w
e
sa
y
ab
out
the
t
w
o
n
um
b
ers
x
and(
y
that
satisfy
:
x
×
y
=
0
c
W
e
can
deduce
that
:
a
−
b
=
0
.
W
e
ha
v
e
just
sho
wn
the
uniqueness
of
the
square
ro
ot.
The
previous
definition
b
ecomes
:
Definition
:
Let
a
b
e
a
p
ositiv
e
n
um
b
er.
W
e
call
the
square
ro
ot
of
the
n
um
b
er
a
the
unique
p
ositiv
e
n
um
b
er
suc
h
that
its
square
is
equal
to
a
.
W
e
denote
this
n
um
b
er
a
Study
of
some
algebraic
prop
erties
:
2
a
Calculate
the
square
of
the
n
um
b
er
5
×
3
.
b
Compare
5
×
3
and
√
15
.
c
Using
the
same
metho
d,
sho
w
the
follo
wing
equalit
y
for
an
y
p
ositiv
e
n
um
b
er
a
and
b
:
√
a
×
√
b
=
√
a
×
b
3
a
Simplify
the
follo
wing
calculation
:
7
3
×
7
3
b
Complete
the
follo
wing
sen
tence
:
7
3
is
a
n
um
b
er
whose
square
is
.
.
.
.
.
.
c
Similarly
,
sho
w
that
for
all
p
ositiv
e
n
um
b
ers
a
and
b
,
with
b
=
0
,
w
e
ha
v
e
a
b
=
a
b
Some
simplifications
of
fractions
:
4
a
What
n
um
b
er
squared
equals
5
2
?
b
Deduce
the
v
alue
of
5
2
.
c
Similarly
,
for
an
y
p
ositiv
e
real
n
um
b
er
a
,
determine
the
v
alue
of
a
2
.
5
a
Establish
the
equalit
y
50
=
5
2
b
y
squaring
eac
h
h
ttps:/
/c
hingmath.fr
sacados/1951
sacados/1345
sacados/1956
E
F
G
mem
b
er.
(Establish
the
e
quality
by
using
the
e
quality)
b
Determine
t
w
o
in
tegers
a
and
b
suc
h
that
:
50
=
a
2
×
b
c
Using
the
questions
2
et
4
,
establish
the
equalit
y
in
another
w
a
y
:
50
=
5
2
3.
The
GPS
system
and
triangulation
E.1344
The
only
instrumen
ts
allo
w
ed
are
a
ruler
and
a
compass
:
1
In
the
figure
b
elo
w,
place
p
oin
t
M
satisfying
the
follo
w-
ing
conditions
:
E
M
=
5
cm
;
F
M
=
4
cm
;
GM
=
7
cm
2
Are
t
w
o
p
oin
ts
sufficien
t
to
p
erfectly
lo
cate
the
p
osition
of
a
third
p
oin
t
in
the
plane?
Justify
y
our
answ
er.
Note
:
la
T
riangulation
b
y
measuring
distances
(or
trilater
ation)
allo
ws
us
to
determine
the
exact
p
osition
of
a
p
oin
t
b
y
kno
wing
its
distance
from
other
p
oin
ts.
On
the
plane,
on
the
surface
of
the
earth,
at
least
three
p
oin
ts
are
needed
to
lo
cate
a
p
oin
t
exactly
.
E.1346
Note
:
Quartz
is
a
mineral
used
to
regulate
the
timing
of
all
mo
dern
w
atc
hes.
Quartz
is
obtained
from
silica,
whic
h
has
piezo
electric
prop
erties
disco
v
ered
b
y
Pierre
and
Jacques
Curie
in
1880.
Here
is
a
brief
description
of
the
piezo
elec-
tric
effect
:
When
sub
jected
to
mec
hanical
stress
(pr
essur
e,
str
etch-
ing)
,
it
dev
elops
an
electric
field
;
When
sub
jected
to
an
electric
field,
quartz
undergo
es
deformation.
The
first
prop
ert
y
is
used
in
particular
in
gas
ligh
ters
:
when
a
button
is
pressed,
the
gas
ligh
ter
creates
an
electric
arc.
The
second
prop
ert
y
is
used
in
mini-motors
(nanote
chnol-
o
gy)
.
In
w
atc
hes,
an
electronic
system
allo
ws
these
t
w
o
prop
erties
to
alternate
and
stabilizes
this
alternation
at
a
frequency
of
32
768
H
z
:
the
quartz
will
th
us
vibrate
regularly
32
768
times
p
er
second.
Let
us
assume
four
ob
jects
with
in
ternal
clo
c
ks
set
to
the
same
time
:
3
transmitters
(
E
,
F
,
G
)
and
1
receiv
er
(
R
)
.
It
is
assumed
that
the
w
a
v
es
emitted
b
y
the
3
transmitters
propagate
at
the
sp
eed
of
ligh
t,
whic
h
is
3
×
10
8
m
=
s
(in
fact
299
792
458
m
=
s
)
.
The
receiv
er
R
is
set
to
the
same
time,
so
it
can
measure
the
time
tak
en
b
y
the
signal
from
eac
h
transmitter.
Here
is
one
of
its
readings
:
Time
of
parcours
the
signal
Distance
actual
(in
cm
)
Distance
to
the
scale
(in
cm
)
from
E
to
R
0
;
0004
s
from
F
to
R
0
;
001
s
from
G
to
R
0
;
0017
s
1
Calculate
the
distance
b
et
w
een
p
oin
t
M
and
eac
h
of
the
receiv
ers.
2
Here
are
p
oin
ts
E
,
F
,
G
sho
wn
on
a
map
at
scale
1
=
6
000
000
h
ttps:/
/c
hingmath.fr
chapExoCorrec/1344
sacados/1344
E
F
G
sacados/1346
E
F
G
S
1
S
2
O
R
S
E
F
G
H
I
R
S
E
F
G
H
I
Premier cas
Second cas
S
R
E
F
G
H
I
S
R
E
F
G
H
I
Troisième cas
Quatrième cas
R
S
T
Place
p
oin
t
R
(c
onstruction
lines
must
r
emain
visible)
.
Note
:
the
GPS
p
ositioning
system
(Glob
al
Positioning
Sys-
tem)
consists
of
29
satellites
in
orbit.
Eac
h
of
these
satellites
carries
an
atomic
clo
c
k
(se
e
b
ox)
and
sends
a
signal
at
reg-
ular
in
terv
als
including
the
satellite’s
p
osition
and
the
date
and
time
the
message
w
as
transmitted.
Quartz
clo
c
k
systems
ha
v
e
a
margin
of
error
of
a
few
sec-
onds
p
er
y
ear,
while
atomic
clo
c
ks,
based
on
the
vibrational
prop
erties
of
the
cesium
atom,
ha
v
e
an
accuracy
of
a
few
seconds
p
er
million
y
ears.
An
observ
er
O
lo
cated
on
Earth
receiving
signals
from
t
w
o
satel-
lites
and
able
to
measure
the
dis-
tance
b
et
w
een
themselv
es
and
these
satellites.
This
is
insuf-
ficien
t
to
determine
their
exact
p
osition
:
the
in
tersection
of
t
w
o
spheres
can
b
e
a
circle.
F
or
a
GPS
receiv
er
to
function
prop
erly
,
it
is
necessary
to
receiv
e
signals
from
at
least
4
satellites
to
determine
its
p
o-
sition
on
Earth.
E.1347
Here
are
four
figures
represen
ting
the
same
p
oin
ts
R
and
S
and
a
con
tour
line
b
et
w
een
these
t
w
o
p
oin
ts.
1
Complete
the
follo
wing
table
:
In
eac
h
case,
complete
the
follo
wing
table
:
R
F
S
F
R
F
−
S
F
R
I
S
I
R
I
−
S
I
1
o
2
o
3
o
4
o
Note
:
Eac
h
of
these
curv
es
represen
ts
the
set
of
p
oin
ts
suc
h
that
the
difference
in
distance
b
et
w
een
them
and
p
oin
ts
R
and
S
is
constan
t.
These
curv
es
are
called
h
yp
erb
olas,
and
p
oin
ts
R
and
S
are
called
the
fo
ci
of
the
h
yp
erb
ola.
These
curv
es
are
used
in
particular
to
determine
the
epicen-
ter
of
an
earthquak
e
:
w
e
do
not
kno
w
the
time
of
emission
of
the
seismic
w
a
v
es,
but
w
e
can
measure
the
difference
in
the
arriv
al
time
of
the
signal
b
et
w
een
t
w
o
receiv
ers.
In
this
case,
three
receiv
ers
are
also
needed
to
determine
the
exact
p
osition
of
the
emission
source.
3
Belo
w
is
a
cross-section
of
the
earth,
where
p
oin
ts
R
,
S
,
and
T
represen
t
receiv
ers
placed
on
the
earth’s
surface
:
W
e
kno
w
that
:
h
ttps:/
/c
hingmath.fr
E
F
G
S
1
S
2
O
sacados/1347
R
S
E
F
G
H
I
R
S
E
F
G
H
I
Premier cas
Second cas
S
R
E
F
G
H
I
S
R
E
F
G
H
I
Troisième cas
Quatrième cas
R
S
T
s
s
s
s
s
s
s
s
s
s
s
S
T
Agde
Sète
Mon
tp
ellier
Nîmes
Balaruc
D31
D31
A9
27
k
m
Balaruc
A9
90
k
m
Nîmes
D31
54
k
m
Nîmes
D31
A9
Balaruc
14
k
m
1
2
3
A
B
C
D
E
F
Nom
bre
de
demi-journées
T
arif
A
T
arif
B
1
8
35
2
16
40
3
4
5
the
seismic
w
a
v
es
arriv
ed
at
receiv
ers
R
and
S
at
the
same
time
while
the
signal
arriv
ed
Δ=4
s
seconds
earlier
at
re-
ceiv
er
S
than
at
receiv
er
T
.
Determine
the
exact
lo
cation
of
the
epicen
ter
F
of
the
earthquak
e.
4.
Unclassifie
d
exer
cises
E.6336
P
aul
w
an
ts
to
go
from
Agde
to
Mon
tp
el-
lier.
He
has
t
w
o
c
hoices
:
either
he
tak
es
the
high
w
a
y
A9
,
or
he
tak
es
the
departmen
tal
road
D31
.
Here,
diagrammed,
is
the
information
he
retriev
es
from
a
map
:
1
He
estimates
that
he
is
a
v
eraging
110
k
m
=
h
on
the
high
w
a
y
and
75
k
m
=
h
on
coun
t
y
roads.
Whic
h
route
should
he
tak
e
to
get
to
the
cit
y
of
Mon
tp
e-
lier
the
fastest?
2
On
the
same
route,
A
deline
lea
v
es
at
9
h
using
the
coun
t
y
road
while
juliette
lea
v
es
at
9
h
15
.
That
she
is
the
driv
er
who
will
arriv
e
in
Mon
tp
elier
the
so
onest.
E.8945
An
asso
ciation
offers
a
v
ariet
y
of
activities
to
k
eep
c
hildren
o
ccupied
during
the
sc
ho
ol
v
aca-
tions.
Sev
eral
rates
are
offered
:
Rate
A
:
8
e
p
er
half
da
y
;
Rate
B
:
a
mem
b
ership
of
30
e
en
titling
the
mem
b
er
to
a
discoun
ted
rate
of
5
e
p
er
half
da
y
.
A
spreadsheet
file
has
b
een
prepared
to
calculate
the
cost
to
b
e
paid
based
on
the
n
um
b
er
of
half
da
ys
of
activities
for
eac
h
of
the
prop
osed
rates
:
Questions
1
,
2
,
4
,
and
5
do
not
require
justification.
1
Complete
this
table.
2
Find
from
the
follo
wing
answ
ers
the
form
ula
that
w
as
en
tered
in
cell
B3
b
efore
stretc
hing
it
to
the
righ
t
:
a
=8
×
B1
b
=30
×
B1+5
c
=5
×
B1+30
×
BA
d
=30+5
×
B1
e
=35
h
ttps:/
/c
hingmath.fr
s
s
s
s
s
s
s
s
s
s
s
S
T
sacados/6336
Agde
Sète
Mon
tp
ellier
Nîmes
Balaruc
D31
D31
A9
27
k
m
Balaruc
A9
90
k
m
Nîmes
D31
54
k
m
Nîmes
D31
A9
Balaruc
14
k
m
chapExoCorrec/8945
sacados/8945
1
2
3
A
B
C
D
E
F
Nom
bre
de
demi-journées
T
arif
A
T
arif
B
1
8
35
2
16
40
3
4
5
quand
est cliqué
cacher
mettre la taille du stylo à
1
e∑acer tout
aller à x:
0
y:
0
Rectangle
ajouter
40
à
x
ajouter
-20
à
y
répéter
5
fois
de∏nir rectangle
stylo en position d’écriture
s’orienter à
90
degrés
avancer de
40
tourner
de
90
degrés
avancer de
20
tourner
de
90
degrés
répéter
2
fois
relever le stylo
ajouter
1
à la taille du stylo
E.8898
W
e
w
an
t
to
mak
e
a
frieze
com-
p
osed
of
rectangles.
T
o
do
this,
w
e
wrote
the
program
b
elo
w
:
Recall
that
the
instruction
ˇ
orien
t
to
90
ı
is
to
orien
t
horizon-
tally
to
the
righ
t.
In
this
exercise,
no
justification
is
required.
1
What
are
the
co
ordinates
of
the
starting
p
oin
t
of
the
plot?
2
Ho
w
man
y
rectangles
are
dra
wn
b
y
the
main
script?
3
Dra
w
the
figure
obtained
with
the
main
script
freehand.
4
a
Without
mo
difying
the
main
script,
w
e
obtained
the
figure
b
elo
w
comp
osed
of
rectangles
of
length
40
pix-
els
and
width
20
pixels.
Prop
ose
a
mo
dification
to
the
blo
c
k
ˇ
rectangle
ı
that
w
ould
result
in
this
figure.
b
Where
can
w
e
then
add
the
instruction
:
in
the
main
script
to
get
the
figure
b
elo
w?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8898
sacados/8898
quand
est cliqué
cacher
mettre la taille du stylo à
1
e∑acer tout
aller à x:
0
y:
0
Rectangle
ajouter
40
à
x
ajouter
-20
à
y
répéter
5
fois
de∏nir rectangle
stylo en position d’écriture
s’orienter à
90
degrés
avancer de
40
tourner
de
90
degrés
avancer de
20
tourner
de
90
degrés
répéter
2
fois
relever le stylo
ajouter
1
à la taille du stylo
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