Question

In a $\Delta ABC,A = \frac{2\pi}{3},b - c = 3\sqrt{3}$ and ar$R = \frac{5}{2} = 2.5\text{ unit}$ Then a is

• A. $6\sqrt{3}cm.$
• B. $9cm.$
• C. 18 cm.
• D. None of these

Answer 1

Answer: (B)

$9cm.$

Answer 2

$\frac{1}{2}bc\sin\frac{2\pi}{3} = \frac{9\sqrt{3}}{2}$ or $\frac{1}{2}.\frac{\sqrt{3}}{2}$.bc =$\frac{9\sqrt{3}}{2}$ ⇒ bc =18

Also, $\cos\frac{2\pi}{3} = \frac{b^{2} + c^{2} - a^{2}}{2bc}$ ⇒ $- \frac{1}{2} = \frac{(b - c)^{2} + 2bc - a^{2}}{2bc}$ or

$(b - c)^{2} + 3bc - a^{2} = 0$ or $27 + 54 = a^{2}$ ⇒ $a = 9$.