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

ABCD is a rhombus.

Perimeter of rhombus = 4 x side

Hence, side AB = BC = CD = DA = $\dfrac{96}{4}$ = 24 cm

∠ ABC = 120°

As we know that diagonal of a rhombus bisect each other at 90°.

In Δ ABO,

∠ ABO = $\dfrac{120°}{2}$ = 60°

$sin 60° = \dfrac{Perpendicular}{Hypotenuse}\\[1em] ⇒ \dfrac{\sqrt3}{2} = \dfrac{AO}{AB}\\[1em] ⇒ \dfrac{\sqrt3}{2} = \dfrac{AO}{24}\\[1em] ⇒ AO = \dfrac{24 \sqrt3}{2}\\[1em] ⇒ AO = 20.78$

∴ AC = 2 x AO = 2 x 20.78 = 41.56 cm

Similarly,

$\text{cos 60°} = \dfrac{Base}{Hypotenuse}\\[1em] ⇒ \dfrac{1}{2} = \dfrac{BO}{AB}\\[1em] ⇒ \dfrac{1}{2} = \dfrac{BO}{24}\\[1em] ⇒ BO = \dfrac{24}{2}\\[1em] ⇒ BO = 12$

∴ BD = 2 x BO = 2 x 12 = 24 cm

Hence, the lengths of the diagonals are: AC = 41.56 cm and BD = 24 cm.