Since all the sides of the figure have the same length, then the figure is a rhombus. Then, its diagonals intersect at an angle of 90°.
Let O be the intersection of the diagonals of the rhombus. Notice that the triangle EOA is a right triangle. Since the side EA is the hypotenuse of the triangle, then, recalling the trigonometric functions:
[tex]\begin{gathered} \cos (30)=\frac{EO}{EA} \\ \sin (30)=\frac{OA}{EA} \end{gathered}[/tex]Use this information to solve for the segments EO and OA:
[tex]\begin{gathered} EO=EA\cdot\cos (30) \\ =10\cdot\frac{\sqrt[]{3}}{2} \\ =5\cdot\sqrt[]{3} \end{gathered}[/tex][tex]\begin{gathered} OA=EA\cdot\sin (30) \\ =10\cdot\frac{1}{2} \\ =5 \end{gathered}[/tex]Since the diagonal EM is twice the segment EO and the diagonal BA is twice the segment OA, then the lengths of the diagonals are:
[tex]\begin{gathered} BA=10 \\ EM=10\cdot\sqrt[]{3} \end{gathered}[/tex]Therefore, the answer is:
[tex]10\text{ yards and }10\cdot\sqrt[]{3}\text{ yards}[/tex]