Question

Aria triunghiului ABC=2rad3, m(BAC)=60, iar b+c=7.
Sa se determine: a=?

Answer 1

$A_{{\Delta ABC}} = \frac{b*c*sinA}{2} \Rightarrow 2 \sqrt{3} = \frac{b*c* \frac{ \sqrt{3} }{2} }{2} = \frac{b*c* \sqrt{3} }{4} \Rightarrow b*c = 8 \\ \\ b+c = 7| ^{2} \Rightarrow b^{2} + c^{2} +2bc = 49 \Rightarrow b^{2} + c^{2}=49-16 \Rightarrow \\ b^{2} + c^{2}=33 \\ \\ a^{2} = b^{2} + c^{2} - 2bc*cosA \Rightarrow a^{2} =33-18* \frac{1}{2} \Rightarrow a^{2} = 33-9 \Rightarrow \\ \Rightarrow a^{2} = 24 \Rightarrow a = \sqrt{24} \Rightarrow a = 2 \sqrt{6}$