If the ratio of angles In a triangle is `2 : 5 : 3`, then the value of least angle is

Options

A. `(pi)/(20)`
B. `(pi)/(10)`
C. `(2pi)/(5)`
D. `(pi)/(5)`

Answer

To find the least angle in a triangle where the ratio of the angles is given as \(2 : 5 : 3\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ratio of Angles**:
The angles in the triangle can be represented in terms of a variable \(x\). Let:
- Angle 1 (\(\theta_1\)) = \(2x\)
- Angle 2 (\(\theta_2\)) = \(5x\)
- Angle 3 (\(\theta_3\)) = \(3x\)

2. **Set Up the Equation for the Sum of Angles**:
We know that the sum of the angles in a triangle is always \(180^\circ\). Therefore, we can write the equation:
\[ \theta_1 + \theta_2 + \theta_3 = 180^\circ \]
Substituting the expressions for the angles, we get:
\[ 2x + 5x + 3x = 180^\circ \]

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ 10x = 180^\circ \]

4. **Solve for \(x\)**:
Divide both sides by 10 to find \(x\):
\[ x = \frac{180^\circ}{10} = 18^\circ \]

5. **Calculate Each Angle**:
Now that we have \(x\), we can find the measures of each angle:
- \(\theta_1 = 2x = 2 \times 18^\circ = 36^\circ\)
- \(\theta_2 = 5x = 5 \times 18^\circ = 90^\circ\)
- \(\theta_3 = 3x = 3 \times 18^\circ = 54^\circ\)

6. **Identify the Least Angle**:
The least angle among \(36^\circ\), \(90^\circ\), and \(54^\circ\) is \(36^\circ\).

7. **Convert to Radians**:
To express the least angle in radians, we use the conversion factor \(\frac{\pi}{180}\):
\[ 36^\circ = 36 \times \frac{\pi}{180} = \frac{\pi}{5} \text{ radians} \]

### Final Answer:
The value of the least angle is \(\frac{\pi}{5}\) radians.