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. `3 : 4 : 5`
B. `4 : 5 : 6`
C. `5 : 9 : 13`
D. `5 : 6 : 7`

Explanation

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. **Let the sides of the triangle be in AP**:
Let the sides of the triangle be \( a, b, c \) such that \( a < b < c \). Since they are in AP, 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 \( \theta \), the middle angle be \( \phi \), and the largest angle be \( 2\theta \) (since the greatest angle is double the smallest).

3. **Use the angle sum property**:
The sum of angles in a triangle is \( \pi \):
\[ \theta + \phi + 2\theta = \pi \]
Simplifying this gives:
\[ 3\theta + \phi = \pi \quad \Rightarrow \quad \phi = \pi - 3\theta \]

4. **Apply the Law of Sines**:
According to the Law of Sines:
\[ \frac{a}{\sin \theta} = \frac{b}{\sin(\pi - 3\theta)} = \frac{c}{\sin(2\theta)} \]
Since \( \sin(\pi - x) = \sin x \), we can rewrite it as:
\[ \frac{a}{\sin \theta} = \frac{b}{\sin(3\theta)} = \frac{c}{\sin(2\theta)} \]

5. **Express sides in terms of \( b \)**:
From the Law of Sines, we can express \( a \) and \( c \) in terms of \( b \):
\[ a = k \sin \theta, \quad b = k \sin(3\theta), \quad c = k \sin(2\theta) \]
where \( k \) is a constant.

6. **Set up the equation for sides in AP**:
Since \( a, b, c \) are in AP:
\[ 2b = a + c \]
Substituting the expressions for \( a \), \( b \), and \( c \):
\[ 2(k \sin(3\theta)) = k \sin \theta + k \sin(2\theta) \]
Dividing through by \( k \) (assuming \( k \neq 0 \)):
\[ 2 \sin(3\theta) = \sin \theta + \sin(2\theta) \]

7. **Use the sine angle addition formula**:
Recall that:
\[ \sin(2\theta) = 2 \sin \theta \cos \theta \]
Thus, we can rewrite the equation:
\[ 2 \sin(3\theta) = \sin \theta + 2 \sin \theta \cos \theta \]
Simplifying gives:
\[ 2 \sin(3\theta) = \sin \theta (1 + 2 \cos \theta) \]

8. **Use the triple angle formula**:
The triple angle formula states:
\[ \sin(3\theta) = 3 \sin \theta - 4 \sin^3 \theta \]
Substituting this into our equation:
\[ 2(3 \sin \theta - 4 \sin^3 \theta) = \sin \theta (1 + 2 \cos \theta) \]
Simplifying gives:
\[ 6 \sin \theta - 8 \sin^3 \theta = \sin \theta + 2 \sin \theta \cos \theta \]
Rearranging leads to:
\[ 5 \sin \theta - 8 \sin^3 \theta - 2 \sin \theta \cos \theta = 0 \]

9. **Factor out \( \sin \theta \)**:
Factoring out \( \sin \theta \):
\[ \sin \theta (5 - 8 \sin^2 \theta - 2 \cos \theta) = 0 \]
This gives us one solution \( \sin \theta = 0 \) (not valid for a triangle), so we focus on:
\[ 5 - 8 \sin^2 \theta - 2 \cos \theta = 0 \]

10. **Solve for \( \sin^2 \theta \)**:
Substitute \( \cos^2 \theta = 1 - \sin^2 \theta \) into the equation and solve for \( \sin^2 \theta \) and \( \cos \theta \).

11. **Find the ratio of sides**:
After solving the equations, we find the ratio \( a:b:c \) which simplifies to \( 4:5:6 \).

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