The area of a rhombus is 216 cm² and one of its diagonals measures 24 cm. Find:

(i) the length of the other diagonal,

(ii) the length of each of its sides,

(iii) its perimeter.

(i) Given:

Area of rhombus = 216 cm²

One diagonal = 24 cm

Let 'd' be the other diagonal of rhombus.

$\Rightarrow \text{Area of rhombus } = \dfrac{1}{2} \times \text{product of diagonals} \\[1em] \Rightarrow 216 = \dfrac{1}{2} \times 24 \times d \\[1em] \Rightarrow 216 = 12 \times d \\[1em] \Rightarrow d = \dfrac{216}{12} \\[1em] \Rightarrow d = 18 \text{ cm}.$

Hence the length of the other diagonal = 18 cm.

(ii) The rhombus is shown in the figure below:

Diagonal AC = 24 cm.

The diagonals of a rhombus bisect each other at right angle.

Then, OA = OC = $\dfrac{24}{2}$ = 12 cm

Diagonal, BD = 18 cm

Then, OB = OD = $\dfrac{18}{2}$ = 9 cm

Applying pythagoras theorem for △AOB, we get:

⇒ AB² = OA² + OB²

⇒ AB² = (12)² + (9)²

⇒ AB² = 144 + 81

⇒ AB² = 225

⇒ AB = $\sqrt{225}$

⇒ AB = 15 cm.

Hence, the length of the each of its side = 15 cm.

(iii) Perimeter of rhombus = 4 × side

= 4 × 15

= 60 cm.

Hence, perimeter of the rhombus = 60 cm.