Question

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

Answer 1

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Answer 2

Given :-

Perimeter = 96 cm

Obtuse angle = 120°

To find :-

Length of its diagonals = ?

Solution :-

Perimeter = 96 cm

$\bold{Perimeter~=~4 \times a}$

[ Where a is the side]

4 × a = 96

$\sf{a~=~\dfrac{96}{4}}$

$\sf{a~=~24cm}$

Side = 24 cm each [All sides of a rhombus are equal]

In a rhombus, the diagonals bisect the vertex angles.

∠DCB = 120°

∠BCO = $\sf{\dfrac{120}{2}}$

∠BCO = 60°

The diagonals of a rhombus bisect each other

$\implies{OA~=~OC}$

$\implies{OB~=~OD}$

at right angles.

In Δ BOC

sin 60° = $\sf{\dfrac{P}{H}=\dfrac{OB}{BC}}$

$\sf{ \dfrac{ \sqrt{3} }{2} =\dfrac{OB}{BC}}$

OB = 12√3 cm

For finding the length of diagonal double the side.

BD = 2 × 12√3

= 24√3

The value of √3 is 1.732

= 24 × 1.732

= 41.57 cm

The length of the first diagonal = $\fbox{41.57~cm}$

cos 60° = $\sf{\dfrac{B}{H} = \dfrac{OC}{BC}}$

$\sf{\dfrac{1}{2}=\dfrac{OC}{24}}$

$\sf{OC~=~\dfrac{24}{2}}$

OC = 12 cm

AC = 2 × 12

= 24 cm

The length of its diagonals :-

• 24 cm
• 41.57 cm