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`

Explanation

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step-by-Step Solution:

1. **Identify the Angles in AP**:
The angles of the triangle are in Arithmetic Progression (AP). Let the angles be \( A \), \( A + D \), and \( A + 2D \), where \( A \) is the smallest angle and \( D \) is the common difference.

2. **Given Information**:
We know that the least angle \( A = 30^\circ \).

3. **Sum of Angles in a Triangle**:
The sum of the angles in a triangle is \( 180^\circ \). Therefore, we can write:
\[ A + (A + D) + (A + 2D) = 180^\circ \]
Simplifying this gives:
\[ 3A + 3D = 180^\circ \]
Dividing the entire equation by 3:
\[ A + D = 60^\circ \]

4. **Substituting the Value of A**:
Since \( A = 30^\circ \):
\[ 30^\circ + D = 60^\circ \]
Solving for \( D \):
\[ D = 60^\circ - 30^\circ = 30^\circ \]

5. **Finding the Angles**:
Now we can find the three angles:
- First angle: \( A = 30^\circ \)
- Second angle: \( A + D = 30^\circ + 30^\circ = 60^\circ \)
- Third angle: \( A + 2D = 30^\circ + 2 \times 30^\circ = 30^\circ + 60^\circ = 90^\circ \)

6. **Greatest Angle**:
The greatest angle in the triangle is \( 90^\circ \).

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

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

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