In a rhombus `ABCD, angleA = 60^(@) and AB = 6 cm`. The length of the diagonal BD is equal to

Answer

To find the length of diagonal \( BD \) in rhombus \( ABCD \) where \( \angle A = 60^\circ \) and \( AB = 6 \, \text{cm} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Properties of a Rhombus:**
- In a rhombus, opposite angles are equal and adjacent angles are supplementary.
- Therefore, if \( \angle A = 60^\circ \), then \( \angle C = 60^\circ \) as well.
- The sum of the angles in a quadrilateral is \( 360^\circ \). Thus, \( \angle B + \angle D = 360^\circ - (60^\circ + 60^\circ) = 240^\circ \).

2. **Determine the Angles \( B \) and \( D \):**
- Since \( \angle B \) and \( \angle D \) are equal (as they are opposite angles in a rhombus), we can set \( \angle B = \angle D \).
- Therefore, \( 2\angle B = 240^\circ \) implies \( \angle B = 120^\circ \) and \( \angle D = 120^\circ \).

3. **Analyzing Triangle \( ABD \):**
- In triangle \( ABD \), we have \( \angle A = 60^\circ \) and \( \angle B = 120^\circ \).
- The third angle \( \angle D \) can be calculated as \( 180^\circ - (60^\circ + 120^\circ) = 0^\circ \) (which is not possible). Thus, we need to use the properties of the triangle formed by the diagonals.

4. **Using the Sine Rule in Triangle \( ABD \):**
- Since \( AB = 6 \, \text{cm} \) and \( \angle A = 60^\circ \), we can find the length of diagonal \( BD \) using the sine rule or by recognizing that triangle \( ABD \) can be split into two \( 30-60-90 \) triangles.
- In a \( 30-60-90 \) triangle, the sides are in the ratio \( 1 : \sqrt{3} : 2 \). Here, \( AB \) is the hypotenuse.

5. **Calculate the Length of \( BD \):**
- The length of diagonal \( BD \) can be calculated as:
\[ BD = AB \times \sin(60^\circ) = 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3} \, \text{cm} \]

### Final Answer:
The length of diagonal \( BD \) is \( 3\sqrt{3} \, \text{cm} \).