If the angle A, B and C of a triangle ABC are in A.P and a:b=1: `sqrt(3)` If c=4 cm then the area (in sq. cm) of this triangle is :

Options

A. `4 sqrt3`
B. `2/sqrt3`
C. `2sqrt3`
D. `4/sqrt3`

Answer

To solve the problem step by step, we will follow the given information and apply the necessary mathematical concepts.

### Step 1: Understand the Given Information
We have a triangle ABC where:
- Angles A, B, and C are in Arithmetic Progression (A.P).
- The ratio of sides a:b = 1:√3.
- Side c = 4 cm.

### Step 2: Express Angles in A.P
Since angles A, B, and C are in A.P, we can express them as:
- Let angle B = b.
- Then angle A = b - d and angle C = b + d, where d is the common difference.

### Step 3: Use the Sum of Angles in a Triangle
The sum of angles in a triangle is 180 degrees:
\[ (b - d) + b + (b + d) = 180 \]
This simplifies to:
\[ 3b = 180 \implies b = 60^\circ \]
Thus, angle B = 60 degrees.

### Step 4: Find Angles A and C
Using the value of b:
- Angle A = b - d = 60 - d
- Angle C = b + d = 60 + d

Since A + B + C = 180 degrees, we can substitute B:
\[ (60 - d) + 60 + (60 + d) = 180 \]
This simplifies to:
\[ 180 = 180 \]
This means we can choose d = 30 degrees. Thus:
- Angle A = 30 degrees
- Angle C = 90 degrees

### Step 5: Identify the Triangle Type
Since angle C = 90 degrees, triangle ABC is a right triangle with:
- Angle A = 30 degrees
- Angle B = 60 degrees
- Angle C = 90 degrees

### Step 6: Use the Sine Rule to Find Sides a and b
Using the sine rule:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Given c = 4 cm, we have:
\[ \frac{c}{\sin C} = \frac{4}{1} = 4 \]
Thus:
\[ a = 4 \cdot \sin(30^\circ) = 4 \cdot \frac{1}{2} = 2 \text{ cm} \]
\[ b = 4 \cdot \sin(60^\circ) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \text{ cm} \]

### Step 7: Calculate the Area of the Triangle
The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times a \times b \]
Substituting the values of a and b:
\[ \text{Area} = \frac{1}{2} \times 2 \times 2\sqrt{3} = 2\sqrt{3} \text{ sq. cm} \]

### Final Answer
The area of triangle ABC is \( \boxed{2\sqrt{3}} \) sq. cm.

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