The length of each side of a rhombus is equal to length of side of square whose diagonal is 20 \(\sqrt{2}\) cm. The length of its diagonals is in ratio 3 : 4, then area of their rhombus is?

384 cm²

Diagonal of a square = \(\sqrt {2}\) × side

20\(\sqrt {2}\) = \(\sqrt {2}\) × side

side = 20 cm. = side of the rhombus (given)

Diagonal of rhombus = AC and BD

ATQ, AC = 3a, BD = 4a

In Δ AOB;

⇒AB² = AO² + BO²

⇒ (20)² = (\(\frac{3a}{2}\))² + (\(\frac{4a}{2}\))²

⇒ 400 × 4 = 9 a² + 16a²

⇒ a² =  (\(\frac{1600}{25}\))

⇒ a = (\(\frac{40}{5}\))

⇒ a = 8

Therefore,

Diagonals are ⇒  AC = 3 × 8= 24, BD = 4 × 8 = 32

Area = (\(\frac{1}{2}\)) × AC × BD = (\(\frac{1}{2}\)) × 24 × 32 = 384 cm²