For `Delta ABC, ` if `81 + 144 a ^(4) + 16b ^(4) + 9c ^(4) =144` abc, (where notations have their usual meaning), then :

Options

A. `a gt b gt c`
B. `A lt B lt C`
C. Area of `Delta ABC = (3 sqrt3)/(8)`
D. Triangle ABC is right angled

Answer

To solve the problem, we start with the equation given for triangle ABC:

\[ 81 + 144a^4 + 16b^4 + 9c^4 = 144abc \]

### Step 1: Rearranging the Equation
First, we can rearrange the equation to isolate the terms on one side:

\[ 144a^4 + 16b^4 + 9c^4 - 144abc + 81 = 0 \]

### Step 2: Recognizing Powers
Next, we can rewrite some of the terms to see if we can simplify the expression. Notice that:

- \( 81 = 3^4 \)
- \( 144 = (12)^2 \)
- \( 16 = (4)^2 \)
- \( 9 = (3)^2 \)

Thus, we can express the equation as:

\[ 3^4 + (12a)^4 + (4b)^4 + (3c)^4 = 144abc \]

### Step 3: Applying the AM-GM Inequality
We can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality here. The AM-GM states that for non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean.

We have four terms:

1. \( 3^4 \)
2. \( (12a)^4 \)
3. \( (4b)^4 \)
4. \( (3c)^4 \)

According to AM-GM:

\[ \frac{3^4 + (12a)^4 + (4b)^4 + (3c)^4}{4} \geq \sqrt[4]{3^4 \cdot (12a)^4 \cdot (4b)^4 \cdot (3c)^4} \]

### Step 4: Finding the Geometric Mean
Calculating the geometric mean:

\[ \sqrt[4]{3^4 \cdot (12a)^4 \cdot (4b)^4 \cdot (3c)^4} = 3 \cdot 12a \cdot 4b \cdot 3c \]

### Step 5: Equating AM and GM
Since we have equality in AM-GM when all terms are equal, we set:

\[ 3 = 12a = 4b = 3c \]

### Step 6: Solving for a, b, and c
From \( 12a = 3 \):

\[ a = \frac{3}{12} = \frac{1}{4} \]

From \( 4b = 3 \):

\[ b = \frac{3}{4} \]

From \( 3c = 3 \):

\[ c = 1 \]

### Step 7: Checking the Triangle Inequality
We need to check if the values satisfy the triangle inequality:

1. \( a + b > c \) → \( \frac{1}{4} + \frac{3}{4} > 1 \) (True)
2. \( a + c > b \) → \( \frac{1}{4} + 1 > \frac{3}{4} \) (True)
3. \( b + c > a \) → \( \frac{3}{4} + 1 > \frac{1}{4} \) (True)

All inequalities hold, so \( a, b, c \) can form a triangle.

### Step 8: Determining Angles
The angles opposite to sides \( a, b, c \) can be compared since \( a < b < c \) implies \( \angle A < \angle B < \angle C \).

### Step 9: Area of Triangle
To find the area, we can use Heron's formula or the formula for a right triangle if applicable. Since we have determined the sides, we can check if it is a right triangle using Pythagoras theorem.

### Conclusion
From the calculations, we conclude:

1. The order of sides is \( a < b < c \).
2. The angles follow the same order.
3. The area can be calculated using the appropriate formula.