A $30^\circ$-$60^\circ$-$90^\circ$ triangle is drawn on the exterior of an equilateral triangle so the hypotenuse of the right triangle is one side of the equilateral triangle. If the shorter leg of the right triangle is 6 units, what is the distance between the two vertices that the triangles do not have in common? Express your answer in simplest radical form.

Mason and christian go to the movie theater and purchase refreshments for their friends. mason spends a total of $45.75 on 3 bags of popcorn and 6 drinks. christian spends a total of $71.50 on 6 bags of popcorn and 4 drinks. write a system of equations that can be used to find the price of one bag of popcorn and the price of one drink. using these equations, determine and state the price of a drink, to the nearest cent.