Question

With usual notations if the angles of a triangle are in the ratio 1 : 2 : 3, then their corresponding sides are in the ratio

• A. 1 : 2 : 3
• B. 1 : $\sqrt{3}$ : 3
• C. 2 : $\sqrt{3}$ : 3
• D. 1 : $\sqrt{3}$ : 2

Hint:

This is the classic, highly repeated $30^\circ-60^\circ-90^\circ$ right-angled triangle! For any template $30^\circ-60^\circ-90^\circ$ geometry problem, remember that the side opposite to $30^\circ$ is $1$, opposite to $60^\circ$ is $\sqrt{3}$, and the hypotenuse is $2$. Keeping this template in mind bypasses the Sine Rule calculations entirely!

Answer 1

The Correct Option is D

Step 1: Find the three angles.
The angles are in the ratio $1:2:3$, so call them $x$, $2x$, $3x$. They add up to $180^\circ$, giving $6x=180^\circ$, so $x=30^\circ$. The angles are $30^\circ$, $60^\circ$, $90^\circ$.

Step 2: Use the sine rule.
Sides are proportional to the sines of the opposite angles. So $a:b:c=\sin30^\circ:\sin60^\circ:\sin90^\circ$.

Step 3: Put in the values.
\[ a:b:c=\frac{1}{2}:\frac{\sqrt3}{2}:1 \]

Step 4: Clear the fractions.
Multiply each part by 2. \[ a:b:c=1:\sqrt3:2 \] \[ \boxed{1:\sqrt3:2} \]