Question

In a triangle, if the angles are in the ratio \( 3:2:1 \), then the ratio of its sides is:

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

Hint:

For a triangle with angles in the ratio \( 3 : 2 : 1 \), the sides are in the ratio \( 2 : \sqrt{3} : 1 \) according to the properties of triangles with \( 30^\circ, 60^\circ, 90^\circ \) angles.

Answer 1

The Correct Option is B

In a triangle, if the angles are in the ratio \( 3 : 2 : 1 \), then we can denote the angles of the triangle as \( 3\alpha, 2\alpha, \) and \( \alpha \), where \( \alpha \) is a constant. By the property of the angles of a triangle, we know that the sum of all angles is \( 180^\circ \): \[ 3\alpha + 2\alpha + \alpha = 180^\circ. \] Simplifying the equation: \[ 6\alpha = 180^\circ \quad \Rightarrow \quad \alpha = 30^\circ. \] Thus, the angles of the triangle are: \[ 3\alpha = 90^\circ, \quad 2\alpha = 60^\circ, \quad \alpha = 30^\circ. \] Now, using the property of the sides of a triangle, we know that the sides opposite to the angles of \( 30^\circ, 60^\circ, 90^\circ \) are in the ratio \( 1 : \sqrt{3} : 2 \).
Thus, the ratio of the sides of the triangle is: \[ 2 : \sqrt{3} : 1. \]