Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 10 - Geometry - 10.3 Polygons, Perimeter, and Tessellations - Exercise Set 10.3 - Page 637: 33

Answer

See below

Work Step by Step

(a) A polygon is a two-dimensional figure, which is of two types - a regular polygon and an irregular polygon. A regular polygon is a figure in which all the sides are of the same length. In an irregular polygon, all sides are of different length. A polygon with 3 sides is termed as a triangle. A polygon with 4 sides is termed as rectangle or quadrilateral. A polygon with 5 sides is called pentagon and so on. 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\[{{360}^{o}}\]. Hence,the polygons surround each vertex have sides four which is called square, six sides which are called hexagon and with 12 sides which are called dodecagon (b) The number of angles that come together at each vertex is three in which one angle is formed by the hexagon, one is formed by the square and one is formed by dodecagon. A measure of an angle of a regular hexagon will be determined by dividing the sum of the measures of all angles, which is \[720{}^\circ \] by its sides, i.e., 6. \[\begin{align} & m\measuredangle =\frac{720{}^\circ }{6} \\ & =120{}^\circ \end{align}\] A measure of an angle of a regular square will be determined by dividing the sum of the measures of all angles, which is \[360{}^\circ \] by its sides that is 4. \[\begin{align} & m\angle =\frac{360{}^\circ }{4} \\ & =90{}^\circ \end{align}\] A measure of an angle of a regular dodecagon will be determined by dividing the sum of the measures of all angles, which is \[{{1800}^{o}}\] by its sides, i.e., 12. \[\begin{align} & m\measuredangle =\frac{1,800{}^\circ }{12} \\ & =150{}^\circ \end{align}\] Hence, the measure of the angle at each vertex of a polygon is\[120{}^\circ \],\[90{}^\circ \],\[150{}^\circ \], respectively. (c) 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\[{{360}^{o}}\]. To check whether a tessellation can be created or not add the measurements of the angles as done below: \[\begin{align} & \text{sum of angles}=120{}^\circ +90{}^\circ +150{}^\circ \\ & =360{}^\circ \end{align}\] Hence, the creation of tessellation is possible because the total of measurement of the angles is\[360{}^\circ \].
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