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Standard XII
Maths
Question
The perimeter of a rhombus is 96 cm and the obtuse angle of it is
120
∘
.
Find the length of its smallest diagonal.
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Solution
Verified by Toppr
Let ABCD be the rhombus with perimeter =
96
Let
∠
A
B
C
=
120
∘
Then,
∠
C
D
A
=
∠
A
B
C
=
120
(Opposite angles)
and,
∠
C
=
∠
A
=
x
Sum of angles of rhombus = 360
∠
A
+
∠
B
+
∠
C
+
∠
D
=
360
2
x
=
360
−
240
x
=
60
∘
Now, all sides of rhombus are equal. hence, length of the side =
96
4
=
24
cm
In
△
A
B
C
cos
B
=
A
B
2
+
B
C
2
−
A
C
2
2
×
A
B
×
B
C
cos
120
=
24
2
+
24
2
−
A
C
2
2
×
24
×
24
−
1
2
=
24
2
+
24
2
−
A
C
2
2
×
24
×
24
−
24
2
=
2
×
24
2
−
A
C
2
A
C
=
24
√
3
A
C
=
41.56
cm
In
△
A
B
D
∠
A
=
60
∘
cos
A
=
A
B
2
+
A
D
2
−
B
D
2
2
×
A
B
×
A
D
cos
60
=
24
2
+
24
2
−
B
D
2
2
×
24
×
24
1
2
=
2
×
24
2
−
B
D
2
2
×
24
×
24
24
2
=
2
×
24
2
−
B
D
2
B
D
=
24
cm
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