© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 10 Geometry.

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© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 10 Geometry

© 2010 Pearson Prentice Hall. All rights reserved Polygons, Perimeters, and Tessellations

© 2010 Pearson Prentice Hall. All rights reserved. Objectives 1.Name certain polygons according to the number of sides. 2.Recognize the characteristics of certain quadrilaterals. 3.Solve problems involving a polygon’s perimeter. 4.Find the sum of the measures of a polygon’s angles. 5.Understand tessellations and their angle requirements. 3

© 2010 Pearson Prentice Hall. All rights reserved. Polygons and Perimeter Polygon: Any closed shape in the plane formed by three or more line segments that intersect only at their endpoints. Regular Polygon: Has sides which are all the same length and angles of all the same measure. Perimeter of a Polygon: The sum of the lengths of its sides. 4

© 2010 Pearson Prentice Hall. All rights reserved. Regular Polygons NamePicture Triangle 3 sides Quadrilateral 4 sides Pentagon 5 sides NamePicture Hexagon 6 sides Heptagon 7 sides Octagon 8 sides 5

© 2010 Pearson Prentice Hall. All rights reserved. Types of Quadrilaterals NameCharacteristicsRepresentation ParallelogramQuadrilateral in which both pairs of opposite sides are parallel and have the same measure. Opposite angles have the same measure RhombusParallelogram with all sides having equal lengths. RectangleParallelogram with four right angles. Because a rectangle is a parallelogram, opposite sides are parallel and have the same measure. SquareA rectangle with all sides having equal length. Each angle measures 90 , and the square is a regular quadrilateral. TrapezoidA quadrilateral with exactly one pair of parallel sides. 6

© 2010 Pearson Prentice Hall. All rights reserved. Example 1: An Application of Perimeter Fencing costs $5.25 per foot. Find the cost to enclose the field with fencing. Solution: a.Find the perimeter of the rectangle in yards. 7

© 2010 Pearson Prentice Hall. All rights reserved. Example 1 continued Fencing costs $5.25 per foot. Find the cost to enclose the field with fencing. Solution: b.Convert to feet. c.Multiply by the cost per foot. 8

© 2010 Pearson Prentice Hall. All rights reserved. Example 2: The Sum of the Measures of a Polygon’s Angles The sum of the measures of the angles of a polygon with n sides is (n – 2)180º. Find the sum of the measures of the angles of an octagon. Solution: Using the formula (n − 2)180º with n = 8, the sum of the measures of an octagon’s angles is: (8 − 2)180º = 6 ∙ 180º = 1080 º 9

© 2010 Pearson Prentice Hall. All rights reserved. Tessellations (Tiling) A pattern consisting of the repeated use of the same geometric figures to completely cover a plane, leaving no gaps and no overlaps. The sum of the measures of the angles that come together at each vertex is always 360º. Most restrictive condition in creating tessellations is that just one type of regular polygon may be used. 10

© 2010 Pearson Prentice Hall. All rights reserved. Examples of Tessellations 11

© 2010 Pearson Prentice Hall. All rights reserved. Example 3: Angle Requirements of Tessellations Explain why a tessellation cannot be created using only regular pentagons. Solution: Apply (n − 2)180º to find the measure of each angle of a regular pentagon. Each angle measures 12