Angle Between Diagonals?
Find the acute angle θ between the diagonals of a rectangle with perimeter 2 p and area 1 6 3 p 2 .
If θ = a tan − 1 ( c b ) , where a , b , c are coprime positive integers, then enter the value of a + b + c .
The answer is 6.
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1 solution
A typo first line
N
O
T
1
6
3
∗
p
b
u
t
1
6
3
∗
p
2
.
A different approach is as under with the same diagram above but not for the given condition.
m
+
n
=
p
,
m
∗
n
=
1
6
3
p
2
.
∴
D
i
a
g
o
n
a
l
2
=
m
2
+
n
2
=
(
m
+
n
)
2
−
2
∗
m
∗
n
=
8
5
∗
p
2
.
⟹
H
a
l
f
D
i
a
g
o
n
a
l
,
(
H
D
)
=
2
1
∗
8
5
∗
p
2
.
∴
A
r
e
a
,
4
A
o
f
q
u
a
r
t
e
r
r
e
c
t
a
n
g
l
e
=
2
1
∗
(
H
D
)
2
∗
S
i
n
θ
∴
A
=
4
∗
8
1
∗
8
5
∗
p
2
∗
S
i
n
θ
=
1
6
3
p
2
.
.
.
g
i
v
e
n
.
∴
S
i
n
θ
=
5
3
,
i
m
p
l
i
e
s
T
a
n
θ
=
4
3
.
∴
1
∗
T
a
n
−
1
4
3
.
=
a
∗
T
a
n
−
1
c
b
,
s
i
n
c
e
a
i
s
n
o
t
c
o
−
p
r
i
m
e
+
t
i
v
e
i
n
t
e
g
e
r
.
T
h
o
u
g
h
t
1
∗
T
a
n
−
1
4
3
=
2
∗
T
a
n
−
1
3
1
.
.
Nice Solution ( + 1 ) !