Question

The lengths of diagonals of a rhombus bear the ratio $1:$ $\sqrt{3}$.The angles of the rhombus are

• A. ${30}^0,{60}^0$
• B. ${30}^0,{120}^0$
• C. ${60}^0,{120}^0$
• D. ${90}^0,{120}^0$

Answer 1

Answer: (C)

${60}^0,{120}^0$

Consider a rhombus $ABCD$
It is given that $\frac{AC}{BD}=\frac{1}{\sqrt{3}}$
Let us consider that the diagonals intersect at $O$
In a rhombus diagonals bisect each other at ${90}^0$
Therefore, $\angle ABO={90}^0$

$\Rightarrow \tan \angle ABO=\frac{AO}{BO}$
$\tan \angle ABO=\frac{AC/2}{BD/2}$
$\tan \angle ABO=\frac{AC}{BD}=\frac{1}{\sqrt{3}}$
Therefore, $\angle ABO={30}^0$
Hence, $\angle B=\angle D=2\left({30}^0\right)={60}^0$
Similarly $\tan \angle BAO=\frac{BO}{AO}$
$\tan \angle BAO=\frac{BD/2}{AD/2}$
$\tan \angle BAO=\frac{BD}{AD}=\sqrt{3}$
$\angle BAO={60}^0$
Hence, $\angle A=\angle C=2\left({60}^0\right)={120}^0$
Hence, answer is option C.