The angles of a triangle are in the ratio 1 : 2 : 3, then find the ratio of the corresponding sides.

Correct option is A

Given:
The angles of a triangle are in the ratio 1 : 2 : 3.​​
Formula Used:
If the angles of the triangle are A : B : C,

then the sides opposite to these angles will be in the ratio $$\sin(A) : \sin(B) : \sin(C).$$

The sum of the angles in a triangle is = $$180^\circ.$$​
Solution:
Let the angles be $$\theta, 2\theta, 3\theta.$$​​
Since the sum of the angles in a triangle is always  $$180^\circ:$$​​
​$$\theta + 2\theta + 3\theta = 180^\circ \\ \ \\ 6\theta = 180^\circ \\ \ \\\theta = 30^\circ$$​​
So, the angles are $$30^\circ, 60^\circ,$$ and $$90^\circ.$$​

The sides opposite to these angles will be in the ratio $$\sin(30^\circ) : \sin(60^\circ) : \sin(90^\circ).$$​

​$$\sin(30^\circ) = \frac{1}{2}, \quad \sin(60^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(90^\circ) = 1$$​

So, the ratio of the sides is:

​$$\frac{1}{2} : \frac{\sqrt{3}}{2} : 1$$​​

1 : $$\sqrt{3}$$​ : 2

The ratio of the corresponding sides is 1 : $$\sqrt{3}$$​ : 2.
Thus, the correct option is (a).