Question

In a rhombus ABCD, ∠A = 60°, AB = 10 cm. Find the length of the diagonals.

Answer 1

Given:

In rhombus ABCD, ∠A = 60°, AB = 10 cm.

All sides are equal in a rhombus. Find the lengths of diagonals AC and BD.

Let the diagonals AC and BD intersect at O.

Diagonals of a rhombus bisect each other at right angles, so ∠AOB = 90°.

Let AO = OC = x and BO = OD = y.

In triangle ABO:

AB = 10 cm (side of rhombus),

∠A = 60°,

Right angle at O between diagonals.

\[\cos 60^\circ = \frac{x}{AB}\]   ... [Use the trigonometric ratio cos 60° in triangle ABO.]

\[\frac{1}{2} = \frac{x}{10}\]

x = 5 cm  

= sin 60° = `y/(AB)`

= \[\frac{\sqrt{3}}{2} = \frac{y}{10}\]  

\[y = 5\sqrt{3} \approx 8.66, \text{cm}\]      ... [Use sin 60° in triangle ABO]

Therefore, diagonals AC = 2x = 2 × 5 = 10 cm,

BD = 2y = 2 × 8.66 ≈ 17.32 cm.

Hence, the lengths of the diagonals of the rhombus are AC = 10 cm and BD = `10sqrt3` cm.