Problem 4 · The angle between the trapezoid's diagonals
Half a regular hexagon: each diagonal makes 30° with the base.
Integer answer, at most 4 digitsIn the figure, sides $BC$, $CD$ and $AD$ have the same length and side $AB$ is twice as long as side $BC$; moreover, these two sides are parallel. Find the value of the angle $\alpha$.
Reasoned solution
Key idea: an isosceles trapezoid with double base and three equal sides is exactly half a regular hexagon cut along a diameter.
Let $BC = CD = AD = 1$ and $AB = 2$. If $M$ is the midpoint of $AB$, triangles $ADM$ and $BCM$ are equilateral: every angle in the figure is $60°$ or $120°$.
With coordinates: $A=(0,0)$, $B=(2,0)$, $D=\left(\tfrac{1}{2}, \tfrac{\sqrt{3}}{2}\right)$, $C=\left(\tfrac{3}{2}, \tfrac{\sqrt{3}}{2}\right)$. The slope of $AC$ is $\tfrac{\sqrt{3}/2}{3/2} = \tfrac{1}{\sqrt{3}}$, i.e. $AC$ makes $30°$ with $AB$. By symmetry, $BD$ also makes $30°$ with the base (on the other end).