a
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4
3
2
1
Outside the middle school program
/ Geometry in space
9 exercises (including 7 corrected)
a
Lateral surface
(4 exercices)
Volume expansion and reduction
(4 exercices)
Pyramid, enlargement, reduction, affine functions
(1 exercice)
A
B
C
D
E
F
6
cm
3
;
2
cm
6
;
8
cm
3
cm
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
I
J
A
B
C
D
E
F
G
I
O
I
H
1.
L
ater
al
surfac
e
E.6604
Consider
the
righ
t
prism
AB
C
D
E
F
sho
wn
b
elo
w
:
1
a
What
is
the
nature
of
the
face
D
E
F
?
b
Determine
the
area
of
face
D
F
E
.
2
a
Of
what
natures
are
the
faces
AB
E
D
,
AC
F
D
and
B
C
F
E
?
b
Determine
the
area
of
eac
h
of
these
faces.
3
Giv
e
the
lateral
area
of
this
righ
t
prism.
E.4949
Consider
the
cub
e
AB
C
D
E
F
GH
with
side
4
cm
sho
wn
b
elo
w
:
1
a
Ho
w
man
y
v
ertices
do
es
this
cub
e
ha
v
e?
b
Ho
w
man
y
edges
do
es
this
cub
e
ha
v
e?
c
Ho
w
man
y
faces
do
es
this
cub
e
ha
v
e?
2
Determine
the
v
olume
of
this
cub
e.
3
Determine
the
lateral
area
of
this
cub
e.
E.586
Consider
the
righ
t
blo
c
k
AB
C
D
E
F
GH
;
note
I
and
J
the
resp
ectiv
e
middles
of
the
AB
C
D
and
E
F
GH
faces.
The
follo
wing
measuremen
ts
are
no
w
assumed
to
b
e
kno
wn
:
AB
=
8
cm
;
AD
=
6
cm
;
AE
=
7
cm
and
w
e
admit
that
the
solid
I
D
C
J
H
G
is
a
righ
t
prism
:
1
Calculate
the
v
olume
of
the
prism
I
D
C
J
H
G
.
2
Calculate
the
lateral
surface
area
of
the
prism
I
D
C
J
H
G
.
E.9123
The
v
olume
of
the
parallelepip
ed
has
b
een
m
ultiplied
b
y
27.
By
ho
w
m
uc
h
ha
v
e
its
dimensions
b
een
en-
larged?
And
its
lateral
area?
2.
V
olume
exp
ansion
and
r
e
duction
E.5461
Consider
a
p
yramid
AB
C
D
O
with
a
square
base
AB
C
D
sho
wn
opp
osite.
Let
I
b
e
the
fo
ot
of
the
p
yramid’s
heigh
t
and
E
the
mid-
p
oin
t
of
[
O
A
]
.
W
e
ha
v
e
the
dimensions
:
AB
=
3
cm
;
I
O
=
4
cm
The
plane
parallel
to
the
base
passing
through
p
oin
t
E
in
ter-
sects.
the
p
yramid,
forming
quadrilateral
E
F
GH
.
1
a
What
is
the
nature
of
the
quadrilateral
E
F
GH
?
b
Justify
that
p
oin
t
F
is
the
midp
oin
t
of
segmen
t
[
O
B
]
.
c
What
can
b
e
said
ab
out
p
yramid
E
F
GH
O
in
relation
to
p
yramid
AB
C
D
O
?
2
a
Determine
the
length
of
segmen
t
[
E
F
]
.
b
Giv
e
the
area
A
of
the
base
AB
C
D
and
the
area
A
of
the
v
ase
E
F
GH
.
c
Giv
e
the
v
alue
of
the
ratio
A
A
of
the
areas.
3
The
form
ula
for
a
p
yramid
is
giv
en
b
y
the
form
ula
:
V
=
1
3
×A
B
×
h
It
is
assumed
that
:
O
I
=
2
cm
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6604
sacados/6604
A
B
C
D
E
F
6
cm
3
;
2
cm
6
;
8
cm
3
cm
chapExoCorrec/4949
sacados/4949
A
B
C
D
E
F
G
H
chapExoCorrec/586
sacados/586
A
B
C
D
E
F
G
H
I
J
chapExoCorrec/9123
sacados/9123
chapExoCorrec/5461
sacados/5461
A
B
C
D
E
F
G
I
O
I
H
A
B
C
D
S
S
A
D
C
B
A
B
C
D
S
A
D
C
B
O
A
D
C
B
O
A
B
C
D
H
S
A
B
C
D
H
S
M
N
O
P
a
Giv
e
the
v
olume
V
of
the
p
yramid
AB
C
D
O
and
the
v
olume
V
of
the
p
yramid
E
F
GH
O
.
b
Giv
e
the
v
alue
of
the
ratio
V
V
of
the
v
olumes.
E.6299
P
aul
visiting
P
aris
admires
the
Pyramid,
made
of
laminated
glass
in
the
cen
ter
of
the
Louvre’s
inner
court
y
ard.
This
regular
p
yramid
has
:
base
a
square
AB
C
D
side
35
meters
;
for
heigh
t
the
segmen
t
[
S
O
]
of
length
22
meters.
P
aul
enjo
y
ed
the
p
yramid
so
m
uc
h
that
as
a
souv
enir
of
his
visit
he
b
ough
t
an
oil
lamp
whose
glass
reserv
oir
is
a
scale
reduction
1
500
of
the
real
p
yramid.
The
lamp’s
instructions
state
that,
once
lit,
it
burns
4
cm
3
of
oil
p
er
hour.
After
ho
w
long
will
there
b
e
no
oil
left
in
the
reserv
oir?
Round
off
to
the
nearest
hour.
Reminder
:
V
olume
of
a
p
yramid
=
one-third
of
the
pro
duct
of
the
area
of
the
base
times
the
heigh
t.
Sho
w
the
approac
h
used
on
the
cop
y
.
An
y
trace
of
researc
h
will
b
e
tak
en
in
to
accoun
t
during
assessmen
t
ev
en
if
the
w
ork
is
not
completely
finished.
E.987
An
hourglass
consists
of
t
w
o
sup
er-
imp
osed
p
yramids,
as
sho
wn
in
the
sk
etc
h
b
elo
w.
The
sand
flo
ws
at
p
oin
t
S
.
The
surface
of
the
sand
is
represen
ted
b
y
the
plane
A
B
C
D
horizon
tal
and
parallel
to
the
bases
of
the
p
yramids
S
AB
C
D
The
p
yramid
S
AB
C
D
is
regular,
its
base
is
a
square
AB
C
D
,
recall
that
the
heigh
t
(
S
O
)
is
p
erp
endicular
to
the
plane
AB
C
D
.
The
sand
lev
el
is
mark
ed
b
y
the
length
S
A
on
the
p
yramid
edge
S
AB
C
D
.
W
e
giv
e
:
O
A
=
27
mm
et
S
O
=
120
mm
.
Throughout
this
problem,
A
is
the
middle
of
[
S
A
]
.
1
Represen
t
the
base
AB
C
D
in
full
size.
2
a
Justify
that
the
triangle
AO
B
is
isosceles
rectangle.
b
Sho
w
that
:
AB
=
27
2
mm
3
a
Calculate
the
area
of
the
square
AB
C
D
b
Deduce
the
v
olume
V
of
the
p
yramid
S
AB
C
D
is
58
320
mm
2
4
The
triangle
S
O
A
is
righ
t-angled.
Sho
w
that
:
S
A
=
123
mm
.
5
Pyramid
S
A
B
C
D
is
a
reduction
of
the
S
AB
C
D
p
yra-
mid.
a
What
can
w
e
sa
y
ab
out
the
straigh
t
lines
(
O
A
)
and
(
O
A
)
?
b
Determine
the
co
efficien
t
of
reduction
S
O
S
O
.
6
Note
V
the
v
olume
of
the
p
yramid
S
A
B
C
D
.
Calculer
V
.
7
It
is
assumed
that
the
v
olume
of
sand
lo
w
ered
is
prop
or-
tional
to
the
time
elapsed.
all
the
sand
runs
out
in
4
min
utes.
After
ho
w
long
is
the
sand
lev
el
in
the
p
osition
studied?
E.5920
A
regular
p
yramid
with
v
ertex
S
has
base
AB
C
D
square
suc
h
that
its
v
olume
V
is
equal
to
108
cm
3
.
Its
heigh
t
measures
9
cm
.
The
v
olume
of
a
p
yramid
is
giv
en
b
y
the
relationship
:
V
olume
of
a
p
yramide
=
aire
of
the
base
×
hauteur
3
1
V
erify
that
the
area
of
AB
C
D
is
36
cm
3
.
Deduce
the
v
alue
of
AB
.
Sho
w
that
the
p
erimeter
of
the
triangle
AB
C
is
equal
to
12+
6
2
cm
.
2
S
M
N
O
P
is
a
reduction
of
the
p
yramid
S
AB
C
D
.
W
e
then
obtain
the
p
yramid
S
M
N
O
P
suc
h
that
the
area
of
the
square
M
N
O
P
is
equal
to
4
cm
2
a
Calculate
the
v
olume
of
the
p
yramid
S
M
N
O
P
.
b
F
or
this
question,
an
y
trace
of
researc
h,
ho
w
ev
er
incomplete,
will
b
e
tak
en
in
to
accoun
t
in
the
ev
aluation
.
Elise
thinks
that
to
obtain
the
p
erimeter
of
the
triangle
M
N
O
,
w
e
simply
divide
the
p
erimeter
of
the
triangle
AB
C
b
y
3
.
Do
y
ou
agree
with
her?
h
ttps:/
/c
hingmath.fr
sacados/6299
A
B
C
D
S
chapExoCorrec/987
sacados/987
Afrique de l'ouest et Asie - juin 2005 - 12 points
S
A
D
C
B
A
B
C
D
S
A
D
C
B
O
A
D
C
B
O
sacados/5920
A
B
C
D
H
S
A
B
C
D
H
S
M
N
O
P
S
A
B
C
D
E
F
G
H
S
A
B
C
D
M
N
P
Q
3.
Pyr
amid,
enlar
gement,
r
e
duction,
affine
functions
E.991
In
the
figure
opp
osite,
S
AB
C
D
is
a
square-based
p
yramid
of
heigh
t
[
S
A
]
suc
h
that
:
AB
=
9
cm
;
S
A
=
12
cm
.
The
triangle
S
AB
is
righ
t-angled
at
A
.
First
part
E
F
GH
is
the
section
of
the
p
yramid
S
AB
C
D
b
y
the
plane
parallel
to
the
base
and
suc
h
that
:
S
E
=
3
cm
.
1
a
Calculate
E
F
.
b
Calculate
S
B
.
2
a
Calculate
the
v
olume
V
of
the
p
yramid
S
AB
C
D
.
b
Giv
e
the
reduction
co
efficien
t
to
go
from
p
yramid
S
AB
C
D
to
p
yramid
S
E
F
GH
.
c
Deduce
the
v
olume
V
of
S
E
F
GH
.
A
v
alue
rounded
to
unit
y
will
b
e
giv
en.
Second
part
Let
M
b
e
a
p
oin
t
of
[
S
A
]
suc
h
that
S
M
=
x
cm
,
where
x
is
b
et
w
een
0
and
12.
W
e
call
M
N
P
Q
the
section
of
the
p
yramid
S
AB
C
D
b
y
the
plane
parallel
to
the
base
passing
through
M
.
1
Sho
w
that
:
M
N
=
0.75
·
x
2
Let
A
(
x
)
b
e
the
area
of
the
square
M
N
P
Q
as
a
function
of
x
.
Sho
w
that
:
A
(
x
)
=
0.5625
·
x
2
.
3
Complete
the
table
b
elo
w
:
Longueur
x
(en
cm
)
0
2
4
6
8
10
12
Area
A
(
x
)
of
the
square
M
N
P
Q
4
Dra
w
a
reference
frame
where
:
on
the
x-axis,
1
cm
represen
ts
1
unit
;
on
the
ordinate
axis,
1
cm
represen
ts
10
units.
the
x-axis
and
y-axis
are
p
erp
endicular.
Place
in
this
frame
the
p
oin
ts
with
abscissa
x
and
ordi-
nate
A
(
x
)
giv
en
b
y
the
table.
5
Is
the
area
of
M
N
P
Q
prop
ortional
to
the
length
S
M
?
Justify
using
the
graph.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/991
sacados/991
Paris - Juin 2006 - 12 points
S
A
B
C
D
E
F
G
H
S
A
B
C
D
M
N
P
Q
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