The sides of a triangle are in A.P. and its area is `(3)/(5)` th of an equilateral triangle of the same perimeter. Find the greatest angle of the triangle

Explanation

To solve the problem step by step, we will follow the reasoning outlined in the video transcript while providing a clear and structured solution.

### Step 1: Define the sides of the triangle
Let the sides of the triangle be in Arithmetic Progression (A.P.). We can denote the sides as:
- \( a = x - d \)
- \( b = x \)
- \( c = x + d \)

### Step 2: Calculate the semi-perimeter
The semi-perimeter \( S \) of the triangle can be calculated as:
\[ S = \frac{a + b + c}{2} = \frac{(x - d) + x + (x + d)}{2} = \frac{3x}{2} \]

### Step 3: Use Heron's formula to find the area of the triangle
According to Heron's formula, the area \( A \) of the triangle is given by:
\[ A = \sqrt{S(S - a)(S - b)(S - c)} \]
Calculating \( S - a \), \( S - b \), and \( S - c \):
- \( S - a = S - (x - d) = \frac{3x}{2} - (x - d) = \frac{x}{2} + d \)
- \( S - b = S - x = \frac{3x}{2} - x = \frac{x}{2} \)
- \( S - c = S - (x + d) = \frac{3x}{2} - (x + d) = \frac{x}{2} - d \)

Now substituting these values into Heron's formula:
\[ A = \sqrt{\frac{3x}{2} \cdot \left(\frac{x}{2} + d\right) \cdot \frac{x}{2} \cdot \left(\frac{x}{2} - d\right)} \]

### Step 4: Simplify the area expression
The area can be simplified as:
\[ A = \sqrt{\frac{3x}{2} \cdot \frac{x}{2} \cdot \left(\frac{x^2}{4} - d^2\right)} = \sqrt{\frac{3x^3}{8} \cdot \left(\frac{x^2}{4} - d^2\right)} \]

### Step 5: Area of the equilateral triangle
The area of an equilateral triangle with the same perimeter \( P = 3x \) is given by:
\[ A_{eq} = \frac{\sqrt{3}}{4} \cdot \left(\frac{P}{3}\right)^2 = \frac{\sqrt{3}}{4} \cdot \left(\frac{3x}{3}\right)^2 = \frac{\sqrt{3}}{4} \cdot x^2 \]

### Step 6: Set up the area relationship
According to the problem, the area of the triangle is \( \frac{3}{5} \) of the area of the equilateral triangle:
\[ \sqrt{\frac{3x^3}{8} \cdot \left(\frac{x^2}{4} - d^2\right)} = \frac{3}{5} \cdot \frac{\sqrt{3}}{4} \cdot x^2 \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \frac{3x^3}{8} \cdot \left(\frac{x^2}{4} - d^2\right) = \left(\frac{3\sqrt{3}}{20} x^2\right)^2 \]
This simplifies to:
\[ \frac{3x^3}{8} \cdot \left(\frac{x^2}{4} - d^2\right) = \frac{27}{400} x^4 \]

### Step 8: Solve for \( d^2 \)
Rearranging and solving for \( d^2 \):
\[ 3x^3 \cdot \frac{x^2}{4} - 3x^3 d^2 = \frac{27}{400} x^4 \cdot 8 \]
This leads to:
\[ d^2 = \frac{16x^2}{25} \]

### Step 9: Find the sides of the triangle
Using the value of \( d \):
\[ d = \frac{4x}{5} \]
Now substituting \( d \) back into the sides:
- \( a = x - d = x - \frac{4x}{5} = \frac{x}{5} \)
- \( b = x \)
- \( c = x + d = x + \frac{4x}{5} = \frac{9x}{5} \)

### Step 10: Find the greatest angle
The greatest angle is opposite the longest side \( c \). Using the cosine rule:
\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]
Substituting the values:
\[ \cos C = \frac{\left(\frac{x}{5}\right)^2 + x^2 - \left(\frac{9x}{5}\right)^2}{2 \cdot \frac{x}{5} \cdot x} \]
Calculating this gives:
\[ \cos C = \frac{\frac{x^2}{25} + x^2 - \frac{81x^2}{25}}{\frac{2x^2}{5}} = \frac{\frac{-15x^2}{25}}{\frac{2x^2}{5}} = -\frac{3}{2} \]

### Conclusion
This implies:
\[ C = 120^\circ \]

Thus, the greatest angle of the triangle is \( \boxed{120^\circ} \).