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4
3
2
1
Outside the high school program
/ Right-angled triangle and circumscribed circle
7 exercises (100% corrected)
a
1.
Unclassifie
d
financial
ye
ars
E.951
The
diagr
am
has
an
orthonor-
mal
c
o
or
dinate
system
(
O
;
I
;
J
)
.
The
unit
of
length
is
the
c
entimeter.
1
a
Plot
p
oin
t
A
(
5
;
3
)
.
b
Using
the
graph,
find
the
co
ordinates
of
−
→
I
A
.
c
Determine
the
distance
I
A
.
2
Consider
p
oin
t
B
−
1
;
21
.
a
Pro
v
e
that
A
and
B
lie
on
the
circle
with
cen
ter
I
and
radius
5.
b
Dra
w
this
circle
and
lo
cate
p
oin
t
B
.
3
a
Lo
cate
p
oin
t
C
,
the
image
of
p
oin
t
A
under
the
sym-
metry
with
cen
ter
I
.
b
Pro
v
e
that
triangle
AB
C
is
a
righ
t
triangle
at
B
.
E.6483
Consider
the
plane
with
a
reference
frame
(
O
;
I
;
J
)
and
the
circle
C
with
cen
ter
K
(
2
;
−
3
)
and
radius
5
.
1
Justify
that
the
p
oin
t
A
(
6
;
−
6
)
is
a
p
oin
t
of
the
circle
C
2
Consider
the
p
oin
t
B
diametrically
opp
osite
the
p
oin
t
A
in
the
circle
C
.
Determine
the
co
ordinates
of
the
p
oin
t
B
.
3
Let
C
b
e
the
p
oin
t
in
the
plane
with
co
ordinates
−
14
5
;
−
8
5
.
Justify
that
the
triangle
AB
C
is
rectangular
at
C
.
E.2723
Consider
the
plane
pro
vided
with
an
orthonormal
frame
of
reference
(
O
;
I
;
J
)
and
the
circle
C
with
cen
ter
K
(
2
;
−
3
)
and
radius
5
.
1
Justify
that
the
p
oin
t
A
(
6
;
−
6
)
is
a
p
oin
t
on
the
circle
C
2
Consider
the
p
oin
t
B
diametrically
opp
osite
the
p
oin
t
A
in
the
circle
C
.
Determine
the
co
ordinates
of
the
p
oin
t
B
.
3
Let
C
b
e
the
p
oin
t
in
the
plane
of
co
ordinate
−
14
5
;
−
8
5
.
Justify
that
the
triangle
AB
C
is
righ
t-angled
at
C
.
E.922
W
e
place
ourselv
es
in
an
or-
thonormal
frame
of
reference
(
O
;
I
;
J
)
.
Consider
the
follo
w-
ing
three
p
oin
ts
:
A
(
1
;
5
)
;
B
(
−
1
;
3
)
;
K
(
7
;
−
1
)
(Optionally
,
a
landmark
can
b
e
created
and
p
oin
ts
placed
as
the
exercise
progresses)
1
Consider
the
p
oin
t
G
the
middle
of
the
segmen
t
[
B
K
]
.
Determine
the
co
ordinates
of
p
oin
t
G
.
2
Let
R
b
e
the
symmetrical
p
oin
t
of
p
oin
t
A
with
resp
ect
to
p
oin
t
G
.
Determine,
b
y
calculation,
the
co
ordinates
of
the
p
oin
t
R
.
3
Sho
w
that
:
B
K
=
4
5
cm
4
It
is
assumed
that
R
A
=
4
5
cm
.
Sho
w,
without
p
er-
forming
calculations,
that
AB
R
K
is
a
rectangle.
5
Consider
the
circle
(
C
)
of
diameter
[
B
K
]
and
the
p
oin
t
E
of
co
ordinates
−
7
5
;
9
5
;
sho
w
that
the
p
oin
t
E
b
elongs
to
the
circle
C
.
6
Deduce,
without
an
y
calculation,
that
the
triangle
B
E
K
is
righ
t-angled
at
E
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/951
sacados/951
chapExoCorrec/6483
sacados/6483
chapExoCorrec/2723
sacados/2723
chapExoCorrec/922
sacados/922
-3
-2
-1
2
3
4
5
I
-4
-3
-2
-1
2
3
4
J
O
A
B
K
C
D
A
B
M
N
P
C
E.4603
In
the
plane
pro
vided
with
a
refer-
ence
frame
O
;
I
;
J
,
consider
the
three
p
oin
ts
:
A
(
−
2
;
2
)
;
B
(
4
;
−
1
)
;
K
1
;
1
2
Consider
the
circle
C
of
diameter
[
AB
]
.
1
Justify
that
the
circle
C
admits
the
p
oin
t
K
as
its
cen
ter
and
whose
radius
has
the
measure
45
2
.
2
Consider
the
p
oin
t
C
with
co
ordinates
8
5
;
−
14
5
.
a
Justify
that
the
p
oin
t
C
is
a
p
oin
t
on
the
circle
C
.
b
Giv
e
the
nature
of
the
triangle
AB
C
.
Justify
y
our
answ
er.
3
The
straigh
t
line
with
equation
y
=
−
14
5
in
tercepts
the
circle
C
at
p
oin
ts
C
and
D
.
a
Justify
that
the
p
oin
t
D
v
erifies
the
equation
:
x
D
−
1
2
=
9
25
b
Deduce
the
co
ordinates
of
p
oin
t
D
.
E.4594
Consider
the
plane
pro
vided
with
a
O
;
I
;
J
orthonormal
reference
frame.
Consider
the
three
p
oin
ts
:
A
(
−
1
;
4
)
;
B
(
−
3
;
−
2
)
;
C
0
;
1
−
6
1
Demonstrate
that
the
triangle
AB
C
is
rectangular
at
C
.
2
Determine,
to
the
nearest
degree,
the
measure
of
the
an-
gle
B
AC
.
3
a
Without
justification,
determine
the
co
ordinates
of
the
p
oin
t
D
diametrically
opp
osite
the
p
oin
t
C
in
the
circle
of
diameter
[
AB
]
.
b
Sho
w
that
the
quadrilateral
AD
B
C
is
a
rectangle.
E.3037
In
the
plane,
consider
a
semicircle
C
of
diameter
[
AB
]
;
let
M
and
N
b
e
t
w
o
p
oin
ts
of
C
suc
h
that
the
half-lines
[
AM
)
and
[
B
N
)
in
tercept
at
the
p
oin
t
P
:
1
Determine
the
v
alue
of
−
−
→
AM
·
−
−
→
B
M
.
2
Establish
the
follo
wing
equalit
y
:
AB
2
=
AP
×
AM
+
P
B
×
N
B
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4603
sacados/4603
-3
-2
-1
2
3
4
5
I
-4
-3
-2
-1
2
3
4
J
O
A
B
K
C
D
chapExoCorrec/4594
sacados/4594
chapExoCorrec/3037
sacados/3037
A
B
M
N
P
C
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