In triangle ABC, `sinA sin B + sin B sin C + sin C sin A = 9//4 and a = 2`, then the value of `sqrt3 Delta`, where `Delta` is the area of triangle, is _______

Explanation

To solve the problem, we need to find the value of \( \sqrt{3} \Delta \) where \( \Delta \) is the area of triangle ABC, given that:

\[ \sin A \sin B + \sin B \sin C + \sin C \sin A = \frac{9}{4} \]
and \( a = 2 \).

### Step 1: Use the given equation

Starting with the equation:

\[ \sin A \sin B + \sin B \sin C + \sin C \sin A = \frac{9}{4} \]

We can multiply both sides by 4:

\[ 4(\sin A \sin B + \sin B \sin C + \sin C \sin A) = 9 \]

### Step 2: Apply the sine product-to-sum identities

Using the product-to-sum identities, we can express the sine products in terms of cosine:

\[ \sin A \sin B = \frac{1}{2} (\cos(A-B) - \cos(A+B)) \]
\[ \sin B \sin C = \frac{1}{2} (\cos(B-C) - \cos(B+C)) \]
\[ \sin C \sin A = \frac{1}{2} (\cos(C-A) - \cos(C+A)) \]

Substituting these into the equation gives:

\[ 2\left(\cos(A-B) + \cos(B-C) + \cos(C-A) - \left(\cos(A+B) + \cos(B+C) + \cos(C+A)\right)\right) = 9 \]

### Step 3: Analyze the triangle properties

Since \( A + B + C = 180^\circ \), we can express \( \cos(A+B) \) as \( -\cos C \), \( \cos(B+C) \) as \( -\cos A \), and \( \cos(C+A) \) as \( -\cos B \). This leads to:

\[ \cos(A-B) + \cos(B-C) + \cos(C-A) = \frac{9}{2} \]

### Step 4: Use the maximum value of cosine

The maximum value of \( \cos \) is 1. Therefore, for the sum of three cosines to equal \( \frac{9}{2} \), each must equal 1:

\[ \cos(A-B) = 1, \quad \cos(B-C) = 1, \quad \cos(C-A) = 1 \]

This implies:

\[ A = B = C \]

Thus, triangle ABC is equilateral.

### Step 5: Calculate the area of the equilateral triangle

For an equilateral triangle with side length \( a = 2 \), the area \( \Delta \) is given by:

\[ \Delta = \frac{\sqrt{3}}{4} a^2 \]

Substituting \( a = 2 \):

\[ \Delta = \frac{\sqrt{3}}{4} (2^2) = \frac{\sqrt{3}}{4} \cdot 4 = \sqrt{3} \]

### Step 6: Calculate \( \sqrt{3} \Delta \)

Now, we need to find \( \sqrt{3} \Delta \):

\[ \sqrt{3} \Delta = \sqrt{3} \cdot \sqrt{3} = 3 \]

### Final Answer

Thus, the value of \( \sqrt{3} \Delta \) is:

\[ \boxed{3} \]