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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

Let ABCD be the rhombus with perimeter = 96
Let ABC=120
Then, CDA=ABC=120 (Opposite angles)
and, C=A=x
Sum of angles of rhombus = 360
A+B+C+D=360
2x=360240
x=60

Now, all sides of rhombus are equal. hence, length of the side = 964
= 24 cm
In ABC
cosB=AB2+BC2AC22×AB×BC

cos120=242+242AC22×24×24

12=242+242AC22×24×24

242=2×242AC2
AC=243
AC=41.56 cm

In ABD
A=60
cosA=AB2+AD2BD22×AB×AD

cos60=242+242BD22×24×24

12=2×242BD22×24×24

242=2×242BD2
BD=24 cm

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