A square, of each side 2, lies above the x-axis and has one vertex at the origin.

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:

Accepted Answer

Vertex $A$: along $30^\circ$ at distance 2: $x_A=2\cos30^\circ=\sqrt3$.

Vertex $C$: along $30^\circ+90^\circ=120^\circ$ at distance 2: $x_C=2\cos120^\circ=-1$.

Vertex $B$ (diagonal, distance $2\sqrt2$, at $75^\circ$): $x_B=2\sqrt2\cos75^\circ = \sqrt3-1$.

$$\text{Sum} = \sqrt3+(\sqrt3-1)+(-1)+0 = 2\sqrt3-2$$

(including the origin's $x=0$ contribution).

Solutions · 1