What is the ratio of a^{2}:b^{2}:c^{2}?
Consider the following for the next two (02) items that follow : The angles A, B and C of a triangle ABC are in the ratio 3:5:4.
- A. 2:2+\sqrt{3}:3 ✓
- B. 2:2-\sqrt{3}:2
- C. 2:2+\sqrt{3}:2
- D. 2:2-\sqrt{3}:3
Correct Answer: A. 2:2+\sqrt{3}:3
Explanation
From the Sine Rule, the sides are proportional to the sines of the opposite angles. Thus, a^2:b^2:c^2 = \sin^2 45^\circ : \sin^2 75^\circ : \sin^2 60^\circ. This evaluates to \frac{1}{2} : \frac{8+4\sqrt{3}}{16} : \frac{3}{4}, which simplifies to \frac{2}{4} : \frac{2+\sqrt{3}}{4} : \frac{3}{4}. The ratio is 2 : 2+\sqrt{3} : 3.