If the sides of a rhombus are 5 cm each and one diagonal is 8 cm, calculate

(i) the length of the other diagonal, and

(ii) the area of the rhombus.

(i) Let ABCD be a rhombus with AC and BD diagonals and each side = 5 cm.

Let AC = 8 cm.

Since, diagonals of rhombus bisect each other at right angles.

∴ AO = 4 cm.

In right angle triangle AOB

Using Pythagoras theorem,

⇒ AB² = AO² + BO²

⇒ 5² = 4² + BO²

⇒ 25 = 16 + BO²

⇒ BO² = 25 – 16 = 9

⇒ BO = $\sqrt{9}$ = 3 cm.

⇒ BD = 2 × BO = 2 × 3 = 6 cm.

Hence, length of other diagonal = 6 cm.

(ii) Area of rhombus = $\dfrac{1}{2}$ × product of diagonals

= $\dfrac{1}{2}$ × 8 × 6

= 4 × 6

= 24 cm².

Hence, area of rhombus = 24 cm².