Find the length of the sides of a triangle, if its angles are in the ratio 2 : 4 : 6 and its circumradius is 12 cm.
Show answer & explanation
Correct answer: C
Explanation
The correct answer is option 3 i.e. 12 cm, 12 cm, 24 cm.
The ratio of the angles of the triangle = 2 : 4 : 6
Circumradius = 12 cm
A triangle with circumradius R and angles A, B, C, the sides opposite to these angles a, b, c will be a = 2R sin A; b = 2R sin B; and c = 2R sin C
Let the three angles be 2x, 4x, 6x.
So, 2x + 4x + 6x = 180°
⇒ 12x = 180°
⇒ x = 180°/12 = 15°
⇒ A = 2x = 30°
⇒ B = 4x = 60°
⇒ C = 6x = 90°
Now, a = 2 × 12 × sin 30° = 24 × (1/2) = 12 cm
b = 2 × 12 × sin 60° = 24 × (√3/2) = 12√3 cm
c = 2 × 12 × sin 90° = 24 × 1 = 24 cm
Therefore, the lengths of the sides of the triangle are 12 cm, 12√3 cm, and 24 cm.