In the sides of a triangle are in the ratio `1:sqrt(3):2,` then the measure of its greatest angle is (a)`pi/6` (b)`pi/3` (c) `pi/2` (d) `(2pi)/3`

Options

A. `pi/6`
B. `pi/2`
C. `pi/3`
D. `2pi/3`

Explanation

To solve the problem, we need to determine the measure of the greatest angle in a triangle where the sides are in the ratio \(1 : \sqrt{3} : 2\).

### Step-by-Step Solution:

1. **Assign the sides of the triangle**:
Let the sides of the triangle be \(a = x\), \(b = \sqrt{3}x\), and \(c = 2x\), where \(x\) is a positive constant.

**Hint**: Assign variables to the sides based on the given ratio.

2. **Identify the longest side**:
Among the sides \(x\), \(\sqrt{3}x\), and \(2x\), the longest side is \(c = 2x\).

**Hint**: The longest side of a triangle is opposite the greatest angle.

3. **Use the Pythagorean theorem**:
To check if this triangle is a right triangle, we can apply the Pythagorean theorem. For a triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the theorem states:
\[ c^2 = a^2 + b^2 \]
Substituting the values:
\[ (2x)^2 = (x)^2 + (\sqrt{3}x)^2 \]
Simplifying this gives:
\[ 4x^2 = x^2 + 3x^2 \]
\[ 4x^2 = 4x^2 \]
This confirms that the triangle is a right triangle.

**Hint**: Verify if the triangle is right-angled by checking the Pythagorean theorem.

4. **Determine the angle**:
In a right triangle, the greatest angle is always \(90^\circ\) (or \(\frac{\pi}{2}\) radians).

**Hint**: Recall that in a right triangle, the angle opposite the longest side is \(90^\circ\).

5. **Conclusion**:
Therefore, the measure of the greatest angle in the triangle is \(\frac{\pi}{2}\).

**Final Answer**: (c) \(\frac{\pi}{2}\)