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

The correct answer is 1 : $\sqrt{3}$ : 2.

Step 1: Find the measure of each angle of the triangle.
Let the three interior angles of the triangle be A, B, and C.
Given that the angles are in the ratio 1 : 2 : 3, we can write them in terms of a constant x as:

$A=x$

$B=2x$

$C=3x$

According to the angle sum property of a triangle, the sum of all interior angles is equal to 180°:

$A+B+C=180°$

$x+2x+3x=180°$

$6x=180°$

$x=30°$

Now, substituting the value of x to find the individual angle measures:

$A=30°$

$B=2(30°)=60°$

$C=3(30°)=90°$

Step 2: Calculate the ratio of the corresponding sides using the Sine Rule.
The Sine Rule states that in any triangle, the side lengths are proportional to the sines of their opposite angles:

$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$

From this rule, the ratio of the sides a : b : c is:

$a:b:c=sinA:sinB:sinC$

Substitute the angle values obtained in Step 1:

$a:b:c=sin30°:sin60°:sin90°$

The standard trigonometric values are:

$sin30°=\frac{1}{2}$

$sin60°=\frac{\sqrt{3}}{2}$

$sin90°=1$

Substituting these values into the ratio gives:

$a:b:c=\frac{1}{2}:\frac{\sqrt{3}}{2}:1$

To express this ratio in simplified whole number form, multiply each term by 2:

$a:b:c=1:\sqrt{3}:2$

Thus, the ratio of the corresponding sides of the triangle is 1 : $\sqrt{3}$ : 2.