The sides of a triangle are in the ratio `1:sqrt3:2` then the angles of the triangle are in the ratio

Options

A. A)`1:3:5`
B. B)`2:3:4`
C. C)`3:2:1`
D. D)`1:2:3`

Answer

To solve the problem, we need to find the angles of a triangle whose sides are in the ratio \(1 : \sqrt{3} : 2\).

### Step-by-Step Solution:

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

2. **Determine the type of triangle**:
Since the sides are in the ratio \(1 : \sqrt{3} : 2\), we can check if they form a right triangle. According to the Pythagorean theorem, if \(c\) is the longest side, then:
\[ a^2 + b^2 = c^2 \]
Substituting the values:
\[ (1k)^2 + (\sqrt{3}k)^2 = (2k)^2 \]
This simplifies to:
\[ k^2 + 3k^2 = 4k^2 \implies 4k^2 = 4k^2 \]
Thus, the triangle is a right triangle with the right angle opposite the longest side \(c\).

3. **Identify the angles**:
In a right triangle, the angles can be found using the ratios of the sides. We know:
- The angle opposite side \(a\) (1k) will be \(\theta_1\).
- The angle opposite side \(b\) (\(\sqrt{3}k\)) will be \(\theta_2\).
- The angle opposite side \(c\) (2k) will be \(90^\circ\).

4. **Calculate the angles**:
Using trigonometric ratios:
- For \(\theta_1\):
\[ \tan(\theta_1) = \frac{a}{b} = \frac{1}{\sqrt{3}} \implies \theta_1 = 30^\circ \]
- For \(\theta_2\):
\[ \tan(\theta_2) = \frac{b}{a} = \frac{\sqrt{3}}{1} \implies \theta_2 = 60^\circ \]
- The third angle is \(90^\circ\).

5. **Express the angles in ratio**:
The angles of the triangle are \(30^\circ\), \(60^\circ\), and \(90^\circ\). To express these angles in ratio:
\[ 30 : 60 : 90 \]
Simplifying this ratio by dividing each term by 30 gives:
\[ 1 : 2 : 3 \]

### Final Answer:
The angles of the triangle are in the ratio \(1 : 2 : 3\).