Given a quadrilateral $ABCD$ such that $AB = BC = m$ and $∠ABC = ∠ADC = 120^\circ$, what is the length of $BD$?

Question

Given a quadrilateral $ABCD$ such that $AB = BC = m$ and $∠ABC = ∠ADC = 120^\circ$, what is the length of $BD$?

I have tried cosine formula but it didn't work.

Answer 1

By using $\angle ABC = \angle ADC = 120^\circ$, we can draw a circle with center $B$ as the following:

Because we have $arc(ADC) = 120^\circ$ and $arc(AEC) = 240^\circ$ so this circle is valid. Now as you can see, we have radius of the circle $|AB| = |BD| = |BC| = m$.