Finding the length of the side of the equilateral triangle

Question

Here, ABCD is a rectangle, and BC = 3 cm. An Equilateral triangle XYZ is inscribed inside the rectangle as shown in the figure where YE = 2 cm. YE is perpendicular to DC. Calculate the length of the side of the equilateral triangle XYZ.

Answer 1

Hint: Extend $EY$ to meet $AB$ at $F$. Drop a vertical line from $X$, hitting $CD$ at $G$. You now have three right triangles with the hypotenuse being the side of the equilateral triangle.

Answer 2

Consider the reference system with the origin in $Z$ in which $DC$ is the real axis, and let $EZ=a$. Then we have $Y=a+2i$ and:

$$e^{\pi i/6}(a+2i) = X $$ so: $$\Im \left[\left(\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)\cdot\left(a+2i\right)\right]=3, $$ or: $$ 1+\frac{\sqrt{3}}{2}a = 3 $$ so $a = \frac{4}{\sqrt{3}}$, and by the Pythagorean theorem: $$ ZY^2 = a^2 + 4 = \frac{16}{3}+4 = \frac{28}{3} $$ so the side length is $2\sqrt{\frac{7}{3}}$.