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

Options

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

Explanation

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

### Step-by-Step Solution:

1. **Assign Variables to the Sides**:
Let the sides of the triangle be:
- \( a = k \)
- \( b = \sqrt{3}k \)
- \( c = 2k \)
Here, \( k \) is a positive constant.

2. **Use the Cosine Rule to Find Angle A**:
The cosine rule states:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ \cos A = \frac{(\sqrt{3}k)^2 + (2k)^2 - (k)^2}{2(\sqrt{3}k)(2k)} \]
Simplifying this:
\[ \cos A = \frac{3k^2 + 4k^2 - k^2}{4\sqrt{3}k^2} = \frac{6k^2}{4\sqrt{3}k^2} = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \]
Therefore, \( A = \frac{\pi}{6} \) (or \( 30^\circ \)).

3. **Use the Cosine Rule to Find Angle B**:
Now, we find angle \( B \) using the cosine rule:
\[ \cos B = \frac{c^2 + a^2 - b^2}{2ac} \]
Substituting the values:
\[ \cos B = \frac{(2k)^2 + (k)^2 - (\sqrt{3}k)^2}{2(k)(2k)} \]
Simplifying:
\[ \cos B = \frac{4k^2 + k^2 - 3k^2}{4k^2} = \frac{2k^2}{4k^2} = \frac{1}{2} \]
Therefore, \( B = \frac{\pi}{3} \) (or \( 60^\circ \)).

4. **Find Angle C**:
Since the sum of angles in a triangle is \( \pi \):
\[ C = \pi - A - B \]
Substituting the values of \( A \) and \( B \):
\[ C = \pi - \frac{\pi}{6} - \frac{\pi}{3} \]
Converting \( \frac{\pi}{3} \) to sixths:
\[ C = \pi - \frac{\pi}{6} - \frac{2\pi}{6} = \pi - \frac{3\pi}{6} = \pi - \frac{\pi}{2} = \frac{\pi}{2} \]
Therefore, \( C = \frac{\pi}{2} \) (or \( 90^\circ \)).

5. **Determine the Ratio of Angles**:
The angles are:
- \( A = \frac{\pi}{6} \)
- \( B = \frac{\pi}{3} \)
- \( C = \frac{\pi}{2} \)

To find the ratio:
\[ A : B : C = \frac{\pi}{6} : \frac{\pi}{3} : \frac{\pi}{2} \]
Dividing each term by \( \frac{\pi}{6} \):
\[ 1 : 2 : 3 \]

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

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