In a triangle with one angle `(2pi)/(3),` the lengths of the sides form an A.P. If the length of the greatest side is 7 cm, the radius of the circumcircle of the triangle is

Options

A. `(7sqrt3)/(3)cm`
B. `(5 sqrt3)/(3)cm`
C. `(2sqrt3)/(3) cm`
D. `sqrt3 cm `

Answer

To solve the problem step-by-step, we will follow the given information about the triangle and use the properties of triangles and the circumradius formula.

### Step 1: Understand the triangle's properties
We know that one angle of the triangle is \( \frac{2\pi}{3} \) radians (which is 120 degrees), and the sides of the triangle are in arithmetic progression (A.P.). The greatest side is given as 7 cm.

### Step 2: Define the sides of the triangle
Let the sides of the triangle be \( a, b, c \) such that \( a \leq b \leq c \). Since the sides are in A.P., we can express them as:
- \( c = 7 \) (the greatest side)
- Let the common difference be \( d \). Then:
- \( b = 7 - d \)
- \( a = 7 - 2d \)

### Step 3: Use the cosine rule
According to the cosine rule:
\[ c^2 = a^2 + b^2 - 2ab \cos A \]
Substituting \( A = \frac{2\pi}{3} \) (where \( \cos \frac{2\pi}{3} = -\frac{1}{2} \)):
\[ 7^2 = (7 - 2d)^2 + (7 - d)^2 - 2(7 - 2d)(7 - d)(-\frac{1}{2}) \]

### Step 4: Expand and simplify the equation
Expanding both sides:
\[ 49 = (49 - 28d + 4d^2) + (49 - 14d + d^2) + (7 - 2d)(7 - d) \]
The term \( (7 - 2d)(7 - d) \) expands to:
\[ 49 - 7d - 14d + 2d^2 = 49 - 21d + 2d^2 \]
Now substituting back:
\[ 49 = 49 - 28d + 4d^2 + 49 - 14d + d^2 + 49 - 21d + 2d^2 \]
Combining like terms:
\[ 49 = 147 - 63d + 7d^2 \]
Rearranging gives:
\[ 7d^2 - 63d + 98 = 0 \]

### Step 5: Solve the quadratic equation
Dividing by 7:
\[ d^2 - 9d + 14 = 0 \]
Factoring:
\[ (d - 7)(d - 2) = 0 \]
Thus, \( d = 7 \) or \( d = 2 \). Since \( d = 7 \) would make \( a = 7 - 2d = -7 \) (not possible), we take \( d = 2 \).

### Step 6: Find the lengths of the sides
Now substituting \( d = 2 \):
- \( a = 7 - 2(2) = 3 \)
- \( b = 7 - 2 = 5 \)
- \( c = 7 \)

### Step 7: Calculate the area of the triangle
Using Heron's formula:
\[ s = \frac{a + b + c}{2} = \frac{3 + 5 + 7}{2} = 7.5 \]
Area \( \Delta \):
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{7.5(7.5 - 3)(7.5 - 5)(7.5 - 7)} = \sqrt{7.5 \cdot 4.5 \cdot 2.5 \cdot 0.5} \]

### Step 8: Calculate the circumradius
Using the formula for the circumradius \( R \):
\[ R = \frac{abc}{4\Delta} \]
Calculating \( abc = 3 \cdot 5 \cdot 7 = 105 \).

### Step 9: Substitute values to find \( R \)
Now substituting the values into the formula:
\[ R = \frac{105}{4 \Delta} \]
Calculating \( \Delta \) from the area calculation and substituting back will give us the circumradius.

### Final Result
After performing the calculations, we find that the circumradius \( R \) is \( \frac{7\sqrt{3}}{3} \).