Question

A square, of each side $2$, lies above the $x-axis$ and has one vertex at the origin. If one of the sides passing through the origin makes an angle $30^{\circ }$ with the positive direction of the $x$-axis, then the sum of the $x$-coordinates of the vertices of the square is :

• A. $2\sqrt{3}-1$
• B. $2\sqrt{3}-2$
• C. $\sqrt{3}-2$
• D. $\sqrt{3}-1$

Answer 1

Answer: (B)

$\frac{x}{\cos 30^{\circ }}=\frac{y}{\sin 30^{\circ }}=2$
$x=\frac{2\sqrt{3}}{2}=\sqrt{3}$
$y=1$
$\frac{x}{\cos 120^{\circ }}=\frac{y}{\sin 120^{\circ }}=2$
$x=-1,y=\sqrt{3}$
$\frac{x}{\cos 75^{\circ }}=\frac{y}{\sin 75^{\circ }}=2\sqrt{2}$
$x=\sqrt{3}-1$
$y=\sqrt{3}+1$
sum $=0+\sqrt{3}+\sqrt{3}-1+(-1)$
$=2\sqrt{3}-2$