Question

$\huge\purple{\mathbb{Question}}$
The perimeter of a rhombus is 96 cm and obtuse angle of it is 120°.Find the lengths of its diagonals​

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Answer 1

The sides of the rhombus must be 96/4 or 24cm.

Draw in the diagonals and you will see that the rhombus comprises four right triangles with angles of 30 and 60 degrees — sides of 12, 24 and 12√3 this gives the shorter diagonal at 24cm and the longer at 24√3cm or ~ 41.57cm.

For better understanding, refer to the above attachment.

Answer 2

Hey mate,

Since in a rhombus all sides are equal.

The diagram is shown above.

Therefore PQ = $\frac{96}{4}$ = 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 = $\frac{1}{2}$ (PQR) = 60°

Sin 60° = $\frac{perp}{hypot}$ = $\frac{PO}{PQ}$ = $\frac{PO}{24}$

But

sin 60° = $\frac{√3}{2}$

$\frac{PO}{24}$ = $\frac{√3}{2}$

PO = 12 √3 = 20.784

Therefore,

PR = 2PO

= 2 x 20.784

= 41.568 cm

Also,

cos 60° = $\frac{base}{hypot}$ = $\frac{OQ}{24}$

But cos° = $\frac{1}{2}$

$\frac{OQ}{24}$ = $\frac{1}{2}$

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.

Hope it helps...

$( Itz ❤Aakanksha❤ here! ) {}^{}$