If the lengths of the sides of a triangle are in AP and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is

Options

A. `1:2:sqrt(2)`
B. `1:sqrt(3):2`
C. `4:5:6`
D. None of these

Answer

To solve the problem, we need to find the ratio of the lengths of the sides of a triangle where the sides are in arithmetic progression (AP) and the greatest angle is double the smallest angle.

### Step-by-Step Solution:

1. **Define the sides of the triangle**:
Let the sides of the triangle be \( A \), \( B \), and \( C \) such that \( A < B < C \). Since they are in AP, we can express them as:
\[ A = B - \lambda, \quad B = B, \quad C = B + \lambda \]
where \( \lambda \) is a positive number.

2. **Use the angle relationship**:
We know that the greatest angle \( C \) is double the smallest angle \( A \):
\[ C = 2A \]

3. **Apply the Law of Sines**:
Using the Law of Sines, we have:
\[ \frac{C}{\sin C} = \frac{A}{\sin A} \]
From the angle relationship, we can also express \( \sin C \) using the double angle formula:
\[ \sin C = \sin(2A) = 2 \sin A \cos A \]

4. **Set up the equation**:
Substituting \( \sin C \) into the Law of Sines gives:
\[ \frac{C}{2 \sin A \cos A} = \frac{A}{\sin A} \]
Simplifying this, we get:
\[ \frac{C}{2 \cos A} = \frac{A}{1} \]
Therefore:
\[ C = 2A \cos A \]

5. **Substituting the expressions for A and C**:
Substitute \( A = B - \lambda \) and \( C = B + \lambda \) into the equation:
\[ B + \lambda = 2(B - \lambda) \cos(B - \lambda) \]

6. **Using the Law of Cosines**:
The cosine of angle \( A \) can be expressed using the Law of Cosines:
\[ \cos A = \frac{B^2 + C^2 - A^2}{2BC} \]
Substitute \( A = B - \lambda \), \( B = B \), and \( C = B + \lambda \) into this equation.

7. **Solve for \( \lambda \)**:
After substituting and simplifying, we find that \( \lambda = \frac{B}{5} \).

8. **Find the lengths of the sides**:
Now substitute \( \lambda \) back into the expressions for \( A \) and \( C \):
\[ A = B - \frac{B}{5} = \frac{4B}{5}, \quad C = B + \frac{B}{5} = \frac{6B}{5} \]

9. **Determine the ratio of the sides**:
The ratio of the sides \( A : B : C \) becomes:
\[ \frac{4B/5}{B} : \frac{B}{B} : \frac{6B/5}{B} = 4 : 5 : 6 \]

### Final Answer:
The ratio of the lengths of the sides of the triangle is \( 4 : 5 : 6 \).

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