Question

the perimeter of a rhombus is 96 cm and obtuse angle of it is 120^ .find the length of its diagonals
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Answer 1

Answer:

hope this helped u dear friend

Answer 2

$\huge{ \underline{ \sf{ \red{Answer:-}}}}$

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 = 360 − 240

➦ x = 60°

Now, all sides of rhombus are equal. hence, length of the side = 96/4 = 24 cm

In △ABC

$\sf{cosB= </p><p>\dfrac{</p><p>AB^{2}</p><p> +BC^{</p><p>2}</p><p> −AC ^{</p><p>2}}{2×AB×BC}}</p><p> </p><p> \\ \\ </p><p> \sf{</p><p></p><p>cos120= </p><p>\dfrac{</p><p>24 ^{</p><p>2}</p><p> +24 ^{</p><p>2}</p><p> −AC ^{</p><p>2}}{2×24×24}</p><p> }</p><p> \\ \\ </p><p> </p><p> \sf{</p><p>\dfrac{−1}{2}</p><p> </p><p> = </p><p>\dfrac{</p><p>24^{ </p><p>2}</p><p> +24 ^{</p><p>2}</p><p> −AC ^{</p><p>2}}{2×24×24}}</p><p> </p><p> </p><p> \\ \\ \sf{</p><p></p><p>−24 ^{</p><p>2}</p><p> =2×24 ^{</p><p>2}</p><p> −AC ^{</p><p>2}</p><p> }</p><p> \\ \\ \sf{</p><p>AC=24 \sqrt{3} </p><p>}</p><p> </p><p> \\ \\ </p><p>\boxed{\frak{ \green{</p><p>AC=41.56 cm}}} \: { \bigstar}</p><p>$

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In △ABD ,, ∠A=60°

$\sf{cosA= </p><p>\dfrac{</p><p>AB ^{</p><p>2}</p><p> +AD ^{</p><p>2}</p><p> −BD ^{</p><p>2}}{2×AB×AD}</p><p> </p><p> }</p><p> </p><p> \\ \\ \sf{</p><p>cos60= </p><p>\dfrac{</p><p>24 ^{</p><p>2}</p><p> +24 ^{</p><p>2}</p><p> −BD ^{</p><p>2}}{2×24×24}</p><p> }</p><p> </p><p> \\ \\ \sf{</p><p></p><p>\dfrac{</p><p>1}{2}</p><p> </p><p> = </p><p>\dfrac{</p><p>2×24 ^{</p><p>2}</p><p> −BD ^{</p><p>2}}{2×24×24}}</p><p> </p><p> </p><p> \\ \\ \sf{ </p><p>24^{2}</p><p> =2×24 ^{</p><p>2}</p><p> −BD ^{</p><p>2}</p><p> }</p><p> \\ \\ \boxed{ \frak{ \green{</p><p>BD=24 cm}}} \:{ \bigstar}$