Answer
A tessellation cannot be created using only regular nine-sided polygons.
Work Step by Step
A tessellation is a type of art that is used to define a relationship between geometry and the visual arts. Tessellations are created by repeated use of same figures that will leave no gap and no overlaps and thus cover the whole plane.
To create a tessellation, the primary requirement is that the sum of the measures of the angles of a regular polygon that are together at each vertex must be\[360{}^\circ \].
To determine whether a tessellation can be created or not, use the formula\[\left( n-2 \right)\times 180{}^\circ \], which computes the sum of the measures of the angle of a regular nonagon and then divides it by the sides of a polygon to find the measure of each angle.
Compute a measure of each angle:
\[\begin{align}
& \text{Measurement of angles}=\frac{\left( n-2 \right)\times 180{}^\circ }{n} \\
& =\frac{\left( 9-2 \right)\times 180{}^\circ }{9} \\
& =\frac{7\times 180}{9} \\
& =140{}^\circ
\end{align}\]
The measure of the angles at each vertex must be \[360{}^\circ \]to form a tessellation. The measure of each angle of a regular nonagon is\[140{}^\circ \]. Hence, the two regular nonagons fill in \[2\times 140{}^\circ =280{}^\circ \]and leave a gap of \[360{}^\circ -280{}^\circ =80{}^\circ \], which does not fulfill the requirement of forming a tessellation by regular nine-sided polygons.
Hence,a tessellation cannot be created using only regular nine-sided polygons.
