ABCD is a rhombus whose side `AB = 4 cm `and `/_ABC = 120^@`, then the length of diagonal BD is equal to :

Options

A. 1 cm
B. 2 cm
C. 3 cm
D. 4 cm

Answer

To find the length of diagonal BD in rhombus ABCD where AB = 4 cm and ∠ABC = 120°, we can follow these steps:

### Step 1: Understand the properties of a rhombus
A rhombus has all sides equal and the diagonals bisect each other at right angles. The diagonals also bisect the angles of the rhombus.

### Step 2: Draw the rhombus and label the angles
Draw rhombus ABCD with AB = 4 cm. Label the angles:
- ∠ABC = 120°
- Since the diagonals bisect the angles, ∠ABD = ∠ABC / 2 = 120° / 2 = 60°.

### Step 3: Find the adjacent angle
The adjacent angle ∠BAD can be found using the property that the sum of adjacent angles in a rhombus is 180°:
- ∠BAD + ∠ABC = 180°
- ∠BAD + 120° = 180°
- ∠BAD = 180° - 120° = 60°.

### Step 4: Analyze triangle ABD
Now, we have triangle ABD where:
- ∠ABD = 60°
- ∠BAD = 60°
- Therefore, ∠ADB = 180° - (60° + 60°) = 60°.

### Step 5: Conclude that triangle ABD is equilateral
Since all angles in triangle ABD are 60°, triangle ABD is an equilateral triangle. In an equilateral triangle, all sides are equal.

### Step 6: Find the length of diagonal BD
Since triangle ABD is equilateral:
- AB = AD = BD
- Given AB = 4 cm, we conclude that BD = 4 cm.

### Final Answer
The length of diagonal BD is **4 cm**.

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