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: Understanding the Question:
We are given the internal angle ratio of a triangle as $1 : 2 : 3$. We need to determine the ratio of its corresponding sides using standard trigonometric laws.

Step 2: Key Formula or Approach:
According to the Sine Rule in trigonometry, the side lengths of any triangle are directly proportional to the sine values of their opposite interior angles: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \implies a : b : c = \sin A : \sin B : \sin C $$

Step 3: Detailed Explanation:
Let the interior angles of the triangle be $x$, $2x$, and $3x$. Since the sum of all angles inside a triangle always equals $180^\circ$: $$ x + 2x + 3x = 180^\circ $$ $$ 6x = 180^\circ \implies x = 30^\circ $$ Substituting $x$ back gives the three individual angles: $$ A = 30^\circ, \quad B = 60^\circ, \quad C = 90^\circ $$ Now, apply the Sine Rule to calculate the side length ratios: $$ a : b : c = \sin(30^\circ) : \sin(60^\circ) : \sin(90^\circ) $$ Substituting the standard exact trigonometric values: $$ a : b : c = \frac{1}{2} : \frac{\sqrt{3}}{2} : 1 $$ Multiply the entire ratio string by 2 to clear out the fractional denominators: $$ a : b : c = 1 : \sqrt{3} : 2 $$

Step 4: Final Answer:
The ratio of the corresponding sides is $1 : \sqrt{3} : 2$, which corresponds to option (D).