If each exterior angle of a regular polygon measures $51\frac{3}{7}\circ$, find the ratio of the number of its diagonals to the number of its sides.

Each exterior angle of a regular polygon is $51\frac{3}{7}^\circ$.

Convert to improper fraction:

$51\frac{3}{7} = \frac{360}{7}$

Number of sides of a regular polygon:

$n = \frac{360}{\text{exterior angle}}$

$n = \frac{360}{\frac{360}{7}} = 7$

Number of diagonals of a polygon:

$D = \frac{n(n-3)}{2}$

$D = \frac{7(7-3)}{2}$

$D = \frac{7 \times 4}{2} = 14$

Ratio of diagonals to sides:

14 : 7 = 2 : 1

Answer: 2 : 1