The perimeter of a rhombus is 96 cm and obtuse angle of it is 120°. Find the lengths of its diagonals.

#### Solution

Since in a rhombus all sides are equal.

The diagram is shown below:

Therefore PQ = `(96)/(4)` = 24 cm, Let ∠ PQR = 120°.

We also know that in rhombus diagonals bisect each other perpendicularly and diagonals bisect the angle at vertex.

Hence POR is a right angle triangle and

POR = `(1)/(2) ("PQR")` = 60°

sin 60° = `"Perp."/"Hypot." = "PO"/"PQ" = "PO"/(24)`

But

sin 60° = `sqrt(3)/(2)`

`"PO"/(24) = sqrt(3)/(2)`

PO = `12sqrt(3)` = 20.784

Therefore,

PR = 2PO

= 2 x 20.784

= 41.568 cm

Also,

cos 60° = `"Base"/"Hypot" = "OQ"/(24)`

But

cos 60° = `(1)/(2)`

`"OQ"/(24) = (1)/(2)`

OQ = 12

Therefore, SQ = 2 x OQ

= 2 x 12

= 24 cm

So, the length of the diagonal PR = 41.568 cm and SQ = 24 cm.