If A Rhombus having `angle A = 60^(@)` . Diagonal BD = 5 am find area and other diagonal .

Answer

To solve the problem of finding the area and the other diagonal of the rhombus given that angle A = 60° and diagonal BD = 5 cm, we can follow these steps:

### Step 1: Understand the properties of a rhombus
A rhombus has the following properties:
- All sides are equal.
- The diagonals bisect each other at right angles (90°).
- The angles opposite each other are equal.

### Step 2: Label the rhombus
Let the rhombus be ABCD where:
- Angle A = 60°
- Diagonal BD = 5 cm
- Let O be the intersection point of the diagonals, which bisects BD.

### Step 3: Calculate the lengths of the segments
Since diagonal BD is bisected at O:
- BO = OD = BD/2 = 5/2 = 2.5 cm

### Step 4: Determine the angles at point O
Since the diagonals bisect each other at right angles:
- Angle AOB = 60° (given)
- Angle AOD = 120° (because angle A + angle C = 180°)

### Step 5: Use trigonometry to find the sides
In triangle AOB:
- Angle AOB = 60°
- Angle OAB = 30° (since triangle AOB is a right triangle)

Using the sine rule in triangle AOB:
- AB = BO / sin(30°)
- Since sin(30°) = 1/2, we have:
\[ AB = 2.5 / (1/2) = 5 \text{ cm} \]

### Step 6: Find AO using cosine
Using the cosine rule in triangle AOB:
\[ AO = BO \cdot \tan(30°) \]
Since tan(30°) = 1/√3:
\[ AO = 2.5 \cdot (1/\sqrt{3}) = \frac{2.5}{\sqrt{3}} \text{ cm} \]

### Step 7: Find the length of diagonal AC
Since the diagonals bisect each other:
\[ AC = 2 \cdot AO = 2 \cdot \frac{2.5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \text{ cm} \]

### Step 8: Calculate the area of the rhombus
The area of a rhombus can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
Where \(d_1\) and \(d_2\) are the lengths of the diagonals.
Here, \(d_1 = BD = 5\) cm and \(d_2 = AC = \frac{5}{\sqrt{3}}\) cm.
So,
\[ \text{Area} = \frac{1}{2} \times 5 \times \frac{5}{\sqrt{3}} = \frac{25}{2\sqrt{3}} \text{ cm}^2 \]

### Final Result
- The other diagonal AC = \( \frac{5}{\sqrt{3}} \) cm
- The area of the rhombus = \( \frac{25}{2\sqrt{3}} \) cm²