The length of the diagonals of a rhombus is in the ratio 4 : 3. If its area is 384 cm², find its side.

Given:

The length of the diagonals of a rhombus is in the ratio 4 : 3.

The area = 384 cm²

Let the length of the diagonals be 4a and 3a.

As we know, the area of rhombus = $\dfrac{1}{2}$ x product of its diagonal

⇒ $\dfrac{1}{2}$ x 4a x 3a = 384

⇒ $\dfrac{1}{2}$ x 12a² = 384

⇒ 6a² = 384

⇒ a² = $\dfrac{384}{6}$

⇒ a² = 64

⇒ a = $\sqrt{64}$

⇒ a = 8

The length of the diagonals be 4a and 3a.

= 4 x 8 cm and 3 x 8 cm

= 32 cm and 24 cm

AC = 32 cm

Then, OA = OC = $\dfrac{32}{2}$ = 16 cm

And, BD = 24 cm

Then, OB = OD = $\dfrac{24}{2}$ = 12 cm

Since the diagonal of a rhombus bisect at 90°.

Applying pythagoras theorem in ΔAOB, we get:

AB² = OA² + OB²

⇒ AB² = (16)² + (12)²

⇒ AB² = 256 + 144

⇒ AB² = 400

⇒ AB = $\sqrt{400}$

⇒ AB = 20 cm

Hence, the length of its side is 20 cm.