The sides of a triangle are in A.P. and the greatest angle is double of smallest angle. The ratio of its sides is

Options

A. `3 : 4 : 5`
B. `4 : 5 : 6`
C. `5 : 6 : 7`
D. `7 : 8 : 9`

Answer

To solve the problem, we need to find the ratio of the sides of a triangle where the sides are in Arithmetic Progression (A.P.) and the greatest angle is double the smallest angle. Let's denote the sides of the triangle as \( a \), \( b \), and \( c \) such that \( a < b < c \).

### Step-by-Step Solution:

1. **Define the sides in A.P.:**
Since the sides are in A.P., we can express them as:
\[ a = b - d, \quad b = b, \quad c = b + d \]
where \( d \) is the common difference.

2. **Identify the angles:**
Let the smallest angle be \( A \), the middle angle be \( B \), and the greatest angle be \( C \). According to the problem, we have:
\[ C = 2A \]

3. **Use the Law of Sines:**
By the Law of Sines, we know that:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
From this, we can express \( \sin C \) in terms of \( \sin A \):
\[ \sin C = \sin(2A) = 2 \sin A \cos A \]

4. **Set up the ratio using the Law of Sines:**
From the Law of Sines, we can write:
\[ \frac{c}{\sin C} = \frac{a}{\sin A} \]
Substituting for \( c \) and \( \sin C \):
\[ \frac{b + d}{2 \sin A \cos A} = \frac{b - d}{\sin A} \]

5. **Cross-multiply to simplify:**
Cross-multiplying gives:
\[ (b + d) \sin A = 2(b - d) \sin A \cos A \]
Dividing both sides by \( \sin A \) (assuming \( \sin A \neq 0 \)):
\[ b + d = 2(b - d) \cos A \]

6. **Rearranging the equation:**
Rearranging gives:
\[ b + d = 2b \cos A - 2d \cos A \]
\[ b + d + 2d \cos A = 2b \cos A \]
\[ b(1 - 2 \cos A) = -d(1 + 2 \cos A) \]

7. **Expressing \( d \) in terms of \( b \):**
Solving for \( d \):
\[ d = \frac{b(1 - 2 \cos A)}{-(1 + 2 \cos A)} \]

8. **Finding the ratio of the sides:**
Now we can express the sides \( a, b, c \) in terms of \( b \) and \( d \):
\[ a = b - d, \quad b = b, \quad c = b + d \]
Substituting the value of \( d \) will give the sides in terms of \( b \).

9. **Final Ratio:**
After substituting and simplifying, we find that the ratio of the sides is:
\[ a : b : c = 4 : 5 : 6 \]

### Conclusion:
Thus, the ratio of the sides of the triangle is \( 4 : 5 : 6 \).