If the circumradius of a triangle is `54/sqrt1463` and the sides are in G.P with common ratio `3/2.` then find the sides of the triangle.

Answer

To solve the problem, we need to find the sides of a triangle given that they are in geometric progression (G.P.) with a common ratio of \( \frac{3}{2} \) and that the circumradius \( R \) is \( \frac{54}{\sqrt{1463}} \).

### Step 1: Define the sides of the triangle
Let the sides of the triangle be \( a, b, c \). Since the sides are in G.P. with a common ratio of \( \frac{3}{2} \), we can express the sides as:
- \( a = x \)
- \( b = \frac{3}{2}x \)
- \( c = \left(\frac{3}{2}\right)^2 x = \frac{9}{4}x \)

### Step 2: Use the circumradius formula
The circumradius \( R \) of a triangle can be expressed in terms of its sides and angles as:
\[ R = \frac{abc}{4K} \]
where \( K \) is the area of the triangle. However, we can also use the formula:
\[ R = \frac{a}{2 \sin A} = \frac{b}{2 \sin B} = \frac{c}{2 \sin C} \]
Given \( R = \frac{54}{\sqrt{1463}} \), we can set up the equations for each side.

### Step 3: Relate the sides to the circumradius
From the circumradius formula, we can write:
\[ \frac{a}{2 \sin A} = \frac{54}{\sqrt{1463}} \]
\[ \frac{b}{2 \sin B} = \frac{54}{\sqrt{1463}} \]
\[ \frac{c}{2 \sin C} = \frac{54}{\sqrt{1463}} \]

### Step 4: Substitute the sides into the circumradius equations
Substituting \( a, b, c \) into the circumradius equations gives:
\[ \frac{x}{2 \sin A} = \frac{54}{\sqrt{1463}} \quad (1) \]
\[ \frac{\frac{3}{2}x}{2 \sin B} = \frac{54}{\sqrt{1463}} \quad (2) \]
\[ \frac{\frac{9}{4}x}{2 \sin C} = \frac{54}{\sqrt{1463}} \quad (3) \]

### Step 5: Solve for \( x \)
From equation (1):
\[ x = \frac{108 \sin A}{\sqrt{1463}} \quad (4) \]
From equation (2):
\[ \frac{3}{2}x = \frac{108 \sin B}{\sqrt{1463}} \quad (5) \]
From equation (3):
\[ \frac{9}{4}x = \frac{108 \sin C}{\sqrt{1463}} \quad (6) \]

### Step 6: Find the relationship between angles
Using the property of triangles, we know that:
\[ \frac{\sin B}{\sin A} = \frac{b}{a} = \frac{\frac{3}{2}x}{x} = \frac{3}{2} \]
\[ \frac{\sin C}{\sin A} = \frac{c}{a} = \frac{\frac{9}{4}x}{x} = \frac{9}{4} \]

### Step 7: Use the sine rule
Using the sine rule, we can express the angles in terms of one angle, say \( A \):
\[ \sin B = \frac{3}{2} \sin A \]
\[ \sin C = \frac{9}{4} \sin A \]

### Step 8: Find the sides
Now substituting back into the expressions for \( x \):
1. From (4): \( x = \frac{108 \sin A}{\sqrt{1463}} \)
2. From (5): \( b = \frac{3}{2}x = \frac{162 \sin A}{\sqrt{1463}} \)
3. From (6): \( c = \frac{9}{4}x = \frac{243 \sin A}{\sqrt{1463}} \)

### Final Step: Calculate the sides
Thus, the sides of the triangle are:
- \( a = \frac{108 \sin A}{\sqrt{1463}} \)
- \( b = \frac{162 \sin A}{\sqrt{1463}} \)
- \( c = \frac{243 \sin A}{\sqrt{1463}} \)