Side lengths appeared in the area too?
I've constructed a quadrilateral with the following properties:
- Side lengths are 6, 7, 8, and 9.
- Area is 6 × 7 × 8 × 9 .
- Interior angles are a ∘ , b ∘ , c ∘ , and d ∘ .
What is the value of a b + b c + c d + d a ?
The answer is 32400.
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1 solution
Call the area of quadrilateral as
K
and
s
be semi-perimeter having sides
w
=
6
,
x
=
7
,
y
=
8
and
z
=
9
.To calculate the area of convex quadrilateral we have
Bretschneider Formula
which is as follow.
K
2
=
(
s
−
w
)
(
s
−
x
)
(
s
−
y
)
(
s
−
z
)
−
w
x
y
z
cos
2
(
2
sum of opp. angles of quad
)
6
.
7
.
8
.
9
=
(
1
5
−
6
)
(
1
5
−
7
)
(
1
5
−
8
)
(
1
5
−
9
)
−
6
.
7
.
8
.
9
cos
2
(
2
a
°
+
c
°
)
s
=
2
w
+
x
+
y
+
z
cos
2
(
2
a
°
+
c
°
)
=
0
⟹
a
°
+
c
°
=
1
8
0
°
Since
a
∘
+
c
∘
=
1
8
0
∘
then
b
∘
+
c
∘
=
1
8
0
∘
too.
∴
a
b
+
b
c
+
c
d
+
a
d
=
(
a
+
c
)
(
b
+
d
)
=
(
1
8
0
)
2
=
3
2
4
0
0
There's a faster way to deal with it.
Hint: What is the maximum area of a quadrilateral with side lengths 6 , 7 , 8 , 9 ?
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Oh!! I see the the area provided is itself maximum which directly implies the sum of opposite sides of quadrilateral is 180°. :)
@Pi Han Goh How do we know when is the area maximum??
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Brahmagupta's formula is a special case of the Bretschneider's formula .
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@Pi Han Goh – @Pi Han Goh Thanks a lot!!! I didn't know about the Bretschneider's formula!!!
Interestingly if you take any cyclic quadrilateral value of ab+bc+cd+ad is always equal to 32,400. Its easy to prove I know., but thats interesting