Chapter 10 - Geometry - 10.3 Polygons, Perimeter, and Tessellations - Exercise Set 10.3 - Page 637: 34

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(a) A polygon is a two-dimensional figure which is of two types that is a regular polygon and an irregular polygon. A regular polygon is a figure in which all the sides are of the same length. In irregular polygon, all sides are of different length. A polygon with three sides is called a triangle.A polygon with four sides is called rectangle or quadrilateral.A polygon with five 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\[360{}^\circ \]. Hence,the polygons surround each vertex have sides four which is called square, six sides which is called hexagon and with three sides which is called Triangle. (b) The number of angles that come together at each vertex is four in which one angle is formed by the hexagon, two angles are formed by the square and one is formed by a triangle. 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 that is 6. \[\begin{align} & m\angle A=\frac{720{}^\circ }{6} \\ & =120{}^\circ \end{align}\] 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. \[\begin{align} & m\angle A=\frac{360{}^\circ }{4} \\ & =90{}^\circ \end{align}\] Hence, two angles from each of the square areoff. A measure of an angle of a regular triangle will be determined by dividing the sum of the measures of all angles which is \[180{}^\circ \] by its sides that is 3. Compute the measurement as follows: \[\begin{align} & m\angle A=\frac{180{}^\circ }{3} \\ & =60{}^\circ \end{align}\] Hence, the number of angles polygon that comes together at each vertex and measure of the angles is 4 and the measurement of the same is\[120{}^\circ \],\[90{}^\circ \],\[90{}^\circ \]\[60{}^\circ \] 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\[360{}^\circ \]. To check whether a tessellation can be created or not, add the measurement of all the angles as follows: \[\begin{align} & \text{Measurement of all the angles}=120{}^\circ +2\times \left( 90{}^\circ \right)+60{}^\circ \\ & =120{}^\circ +180{}^\circ +60{}^\circ \\ & =360{}^\circ \end{align}\] Hence, tessellation can be created as the sum of the measurement of all the four angles are.