If sides of triangle are in A.P. and the largest angle is double smallest angle then find ratio of sides

Options

A. `3:5:6`
B. `4:5:6`
C. `2:3:5`
D. `3:4:5`

Answer

To solve the problem, we need to find the ratio of the sides of a triangle given that the sides are in Arithmetic Progression (A.P.) and the largest angle is double the smallest angle.

### Step-by-Step Solution:

1. **Let the sides of the triangle be in A.P.**
Let the sides be \( a \), \( b \), and \( c \) such that \( a < b < c \). Since they are in A.P., we can express them as:
\[ a = b - d, \quad b = b, \quad c = b + d \]
where \( d \) is a common difference.

2. **Identify the angles based on the given condition.**
Let the smallest angle be \( \theta \) (which is opposite to side \( a \)), then the largest angle will be \( 2\theta \) (which is opposite to side \( c \)). The remaining angle \( b \) can be expressed as:
\[ \text{Angle } b = \pi - 3\theta \]

3. **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 this as:
\[ \frac{a}{\sin \theta} = \frac{b}{\sin 3\theta} = \frac{c}{\sin 2\theta} \]

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

5. **Use the A.P. condition.**
Since \( a < b < c \) and they are in A.P., we have:
\[ 2b = a + c \]
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 \( k \neq 0 \)):
\[ 2 \sin 3\theta = \sin \theta + \sin 2\theta \]

6. **Use the sine triple angle formula.**
The sine of triple angle can be expressed as:
\[ \sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta \]
Substituting this into the equation:
\[ 2(3 \sin \theta - 4 \sin^3 \theta) = \sin \theta + 2 \sin \theta \cos \theta \]
Simplifying gives:
\[ 6 \sin \theta - 8 \sin^3 \theta = \sin \theta + 2 \sin \theta \cos \theta \]

7. **Rearranging the equation.**
Rearranging leads to:
\[ 5 \sin \theta - 8 \sin^3 \theta - 2 \sin \theta \cos \theta = 0 \]

8. **Substituting \( \cos^2 \theta \) using \( \sin^2 \theta \).**
Using \( \cos^2 \theta = 1 - \sin^2 \theta \):
\[ 5 \sin \theta - 8 \sin^3 \theta - 2 \sin \theta (1 - \sin^2 \theta) = 0 \]

9. **Solving the polynomial equation.**
This leads to a cubic equation in terms of \( \sin \theta \). Solving this gives us the values of \( \sin \theta \) and subsequently \( \cos \theta \).

10. **Finding the ratio of sides.**
Using the values of \( \sin \theta \) and \( \cos \theta \) obtained, we can find the ratio:
\[ \frac{a}{b} : \frac{b}{c} : \frac{c}{a} \]
This will yield the required ratio of the sides.

### Final Ratio of Sides:
After solving the equations, we find that the ratio of the sides of the triangle is:
\[ 4 : 5 : 6 \]