Question

In a triangle $ABC,$ if $2a=\sqrt{3}b+c,$ then

• A. $c^2=a^2+b^2$
• B. $a^2=b^2+c^2$
• C. $b^2=a^2+c^2-ac\sqrt{3}$
• D. $a=b=c$

Answer 1

The correct option is B $a^2=b^2+c^2$

$a=\frac{\sqrt{3}}{2}b+\frac{c}{2}$
$\{\text{Comparing with projection formula}a=b\cos C+c\cos B\}$
$\Rightarrow \cos C=\frac{\sqrt{3}}{2},\cos B=\frac{1}{2}$
$\Rightarrow C={30}^{\circ },B={60}^{\circ }$
$\Rightarrow △ABC$ is right angled at $A$
$\therefore a^2=b^2+c^2$