In a triangle ABC, medians AD and BE are deawn. IF `AD= 4, angle D AB=(pi)/(6) and angle ABE=(pi)/(3),` then the area of the trianlge ABC is-

Options

A. `(8)/(3sqrt(3))`
B. `(16)/(3sqrt(3))`
C. `(32)/(3sqrt(3))`
D. `(64)/(3sqrt(3))`

Answer

To find the area of triangle ABC given the medians AD and BE, we can follow the steps below:

### Step 1: Understand the Problem
We have a triangle ABC with medians AD and BE. We know the following:
- Length of median AD = 4
- Angle DAB = π/6
- Angle ABE = π/3

### Step 2: Draw the Triangle
Draw triangle ABC and mark the points:
- A, B, C as the vertices of the triangle.
- D is the midpoint of BC (since AD is a median).
- E is the midpoint of AC (since BE is a median).

### Step 3: Identify Key Points
Since AD is a median, it divides BC into two equal parts:
- Let BD = DC = x (where x is half the length of BC).

### Step 4: Use the Centroid
The centroid O of triangle ABC divides median AD in the ratio 2:1.
- AO = (2/3) * AD = (2/3) * 4 = 8/3
- OD = (1/3) * AD = (1/3) * 4 = 4/3

### Step 5: Use Trigonometry in Triangle ABO
In triangle AOB, we can use the tangent of angle ABE:
- tan(π/3) = AO / BO
- We know tan(π/3) = √3, so:
- √3 = (8/3) / BO
- BO = (8/3) / √3 = (8/3) * (1/√3) = 8/(3√3)

### Step 6: Calculate Area of Triangle ADB
The area of triangle ADB can be calculated using the formula:
- Area = (1/2) * base * height
- Here, base = AD = 4 and height = BO = 8/(3√3):
- Area of triangle ADB = (1/2) * 4 * (8/(3√3))
- Area = 2 * (8/(3√3)) = 16/(3√3)

### Step 7: Calculate Area of Triangle ABC
Since median AD divides triangle ABC into two equal areas:
- Area of triangle ABC = 2 * Area of triangle ADB
- Area of triangle ABC = 2 * (16/(3√3)) = 32/(3√3)

### Final Answer
The area of triangle ABC is \( \frac{32}{3\sqrt{3}} \).

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