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Question

The perimeter of a rhombus is 96 cm and obtuse angle of it is 120,the lengths of its smallest diagonal is ____.

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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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