Question : The length of each side of a rhombus is equal to the length of the side of a square whose diagonal is $40\sqrt2$ cm. If the length of the diagonals of the rhombus is in the ratio $3:4$, then its area ( in cm2) is:
Option 1: 1550
Option 2: 1600
Option 3: 1535
Option 4: 1536
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Correct Answer: 1536
Solution :
The length of each side $(s)$ of the rhombus is equal to the length of the side of a square whose diagonal = $40\sqrt2\operatorname{cm }$
$⇒\sqrt2×s=40\sqrt2 $
$⇒s=40\operatorname{ cm }$
The length of each side of the rhombus is $40\operatorname{ cm }$. The diagonals of the rhombus are in the ratio $3:4$.
Let the diagonals of the rhombus as $3x$ and $4x$.
Since the diagonals of a rhombus bisect each other at right angles, it forms a right triangle with half the length of one diagonal as the base, half the length of the other diagonal as the height, and the side of the rhombus as the hypotenuse.
Using the Pythagorean theorem,
$⇒\left(\frac{3x}{2}\right)^2 + \left(\frac{4x}{2}\right)^2 = 40^2$
$⇒\left(\frac{9x^2}{4}\right) + 4x^2 = 1600$
$⇒\frac{25x^2}{4} = 1600$
$⇒x^2=256$
$⇒x = 16\operatorname{ cm }$
The diagonals of the rhombus are $48\operatorname{ cm }$ and $64\operatorname{ cm }$.
The area of a rhombus = $\frac{1}{2}d_1d_2 = \frac{1}{2} \times 48 \times 64 = 1536\operatorname{ cm^2 }$
Hence, the correct answer is 1536.
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