Question

With the usual notation, in $\Delta ABC$, if $\angle A+\angle B=120^{\circ },a=\sqrt{3}+1$ and $b=\sqrt{3}-1$ , then the ratio $\angle A:\angle B$ , is :

• A. 7 : 1
• B. 5 : 3
• C. 9 : 7
• D. 3 : 1

Answer 1

Answer: (A)

$A+B=120^{\circ }$
$\tan \frac{A-B}{2}=\frac{a-b}{a+b}\cot \left(\frac{C}{2}\right)$
$=\frac{\sqrt{3}+1-\sqrt{3}+1}{2\left(\sqrt{3}\right)}\cot \left(30^{\circ }\right)=\frac{1}{\sqrt{3}}.\sqrt{3}=1$
$\frac{A-B}{2}=45^{\circ }\Rightarrow A-B=90^{\circ }$
$A+B=120^{\circ }$
$------$
$2A=210^{\circ }$
$A=105^{\circ }$
$B=15^{\circ }$