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Chapter 10: Problem 37

Can a tessellation be created using only regular nine-sided polygons? Explain your answer.

Short Answer

Expert verified
No, a tessellation cannot be created using only regular nine-sided polygons because their interior angles do not meet the conditions needed for tessellation.

Step by step solution

01

Understanding the properties of a regular nine-sided polygon (nonagon)

A regular nine-sided polygon, also called a regular nonagon, has interior angles of \(140^\circ\). This can be calculated by the formula: \((n-2)\times 180/n\), where \(n\) is the number of sides. Thus, for a nonagon: \((9-2)\times 180/9 = 140^\circ\).
02

Examining tessellation conditions

For a regular polygon to tessellate, the interior angles must add up to \(360^\circ\) when they meet at a point. So we divide \(360^\circ\) by the interior angle of the regular nonagon, \(140^\circ\). This division yields approximately \(2.57\). Since the number of polygons that can meet at a vertex must be a whole number, it means that regular nonagons cannot tessellate because only an approximate number of \(2.57\) nonagons can meet at a point, which is not possible in reality.
03

Formulating the conclusion

Since regular nine-sided polygons (nonagons) do not meet the conditions for tessellation, it is not possible to create a tessellation using only regular nonagons. Thus, the answer to the question is no.

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