The angles of a triangle are in AP and the least angle is `30^(@)`. What is the greatest angle (in radian) ?

Options

A. `pi/2`
B. `pi/3`
C. `pi/4`
D. `pi`

Answer

To solve the problem, we need to find the greatest angle of a triangle where the angles are in arithmetic progression (AP) and the least angle is given as \(30^\circ\).

### Step-by-step Solution:

1. **Understanding the Angles in AP**:
- Let the angles of the triangle be \(a\), \(b\), and \(c\) where \(a < b < c\).
- Since the angles are in AP, we can express them as:
- \(a = 30^\circ\) (the least angle)
- \(b = a + d\)
- \(c = a + 2d\)
- Here, \(d\) is the common difference.

2. **Using the Sum of Angles in a Triangle**:
- We know that the sum of the angles in a triangle is \(180^\circ\):
\[ a + b + c = 180^\circ \]
- Substituting the expressions for \(b\) and \(c\):
\[ a + (a + d) + (a + 2d) = 180^\circ \]
- Simplifying this gives:
\[ 3a + 3d = 180^\circ \]
- Dividing the entire equation by 3:
\[ a + d = 60^\circ \]

3. **Finding the Common Difference**:
- Since \(a = 30^\circ\):
\[ 30^\circ + d = 60^\circ \]
- Solving for \(d\):
\[ d = 60^\circ - 30^\circ = 30^\circ \]

4. **Calculating the Angles**:
- Now we can find \(b\) and \(c\):
- \(b = a + d = 30^\circ + 30^\circ = 60^\circ\)
- \(c = a + 2d = 30^\circ + 2 \times 30^\circ = 30^\circ + 60^\circ = 90^\circ\)

5. **Finding the Greatest Angle**:
- The greatest angle \(c\) is \(90^\circ\).

6. **Converting to Radians**:
- To convert degrees to radians, we use the conversion factor \(\frac{\pi \text{ radians}}{180^\circ}\):
\[ c = 90^\circ \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians} \]

### Final Answer:
The greatest angle in radians is \(\frac{\pi}{2}\).