A rectangle with dimensions of 24 cm and 28 cm was reconstructed to make a rhombus with the same perimeter as that of the rectangle and $$120^o$$ as one of its angles. The are  of the rhombus was:​

Correct option is D

Given:

Rectangle dimensions: 24 cm and 28 cm.

Rhombus has the same perimeter as the rectangle.

One of the angles of the rhombus is 120^∘

Formula Used:

Perimeter of the rectangle = 2(l + b)

Perimeter of the rhombus = 4 × side

Area of the rhombus = side​²​ × sin(θ), where θ is one of the angles 

Solution:  

​Perimeter of the rectangle = 2(24 + 28) = 2 × 52 = 104 cm

The perimeter of the rhombus is the same as the rectangle, so:

4×side = 104

side = $$\frac{104}4$$ = 26cm

​​From trigonometric values:

​$$sin(120^ ∘ )=sin(180^ ∘ −60 ^∘ )=sin(60^ ∘ )= \frac{\sqrt3}{2}​ ​ .$$ 

So, Area of rhombus;

​$$A = side^2\cdot \sin(\theta) = 26^2 \times \frac{\sqrt{3}}{2} = \frac{676 \sqrt{3}}{2} = 338 \sqrt{3} \, \text{cm}^2$$​​