* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Section 10.3 – Polygons, Perimeter, and Tessellations – pg 126
Survey
Document related concepts
Technical drawing wikipedia , lookup
Penrose tiling wikipedia , lookup
Approximations of π wikipedia , lookup
Rational trigonometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Complex polytope wikipedia , lookup
Regular polytope wikipedia , lookup
Multilateration wikipedia , lookup
Euler angles wikipedia , lookup
List of regular polytopes and compounds wikipedia , lookup
Compass-and-straightedge construction wikipedia , lookup
Euclidean geometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Transcript
126 Section 10.3 – Polygons, Perimeter, and Tessellations Objective #1: Understanding & Classifying Different Types of Polygons. A Polygon is closed two-dimensional geometric figure consisting of at least three line segments for its sides. If all the sides are the same length and all the angles have the same measure, then the figure is called a Regular Polygon. The points where the two sides intersect are called vertices. Triangle 3 sides Quadrilateral 4 sides Pentagon 5 sides Hexagon 6 sides Octagon 8 sides Nonagon 9 sides Decagon 10 sides Dodecagon 12 sides Heptagon 7 sides Regular Polygons Triangle Quadrilateral Pentagon Hexagon Octagon Nonagon Decagon Dodecagon Heptagon 127 Objective #2: Understanding properties of various quadrilaterals. Parallelogram Rectangle A B E F C D Properties: AB || CD AC || BD AB = CD AC = BD m∠A = m∠D m∠B = m∠C Objective #3: G H Properties: EF || GH EG || FH EF = GH EG = FH All the angles measure 90˚. The diagonals EH and GF are equal. Square I Rhombus M N J K L Properties: IK || JL IJ || KL All sides are equal. All the angles measure 90˚. The diagonals IL and KJ are equal. O Trapezoid Q R P S Properties: MN || OP MO || NP All sides are equal. m∠M = m∠P m∠N = m∠O T Properties: Only two sides are parallel: ST || QR Finding the perimeter of a figure. The Perimeter is the length around the outside of a closed two dimensional figure. For a polygon, the perimeter is the sum of the length of the sides of the polygon. We use the idea of perimeter when we calculate how much fencing we need to enclose a garden or the amount of wood we need to frame in a door. Find the perimeter of the following: Ex. 1 4 ft Ex. 2 4 ft 5 ft 9 13 6 ft 6 ft Solution: To find the perimeter, simply add up the lengths of the sides: P = 6 + 4 + 4 + 5 + 6 = 25 ft. € 11 13 m m 12 13 m Solution: € To find the perimeter, simply add up the lengths of the sides: 9 12 11 32 P= m +€ m + m= m. € 13 € 13 € 13 € 13 128 Perimeter of a Triangle In general, if a, b, and c are the lengths of the sides of a triangle, then the formula for the perimeter of a triangle is P = a + b + c. a b c Find the perimeter of the following: Ex. 3 Ex. 4 7.5 m 3.8 m 3.8 m 9 yd 9 yd 7.5 m 9 yd 9 yd Solution: To find the perimeter, simply add up the lengths of the sides: P = 2(7.5) + 2(3.8) = 22.6 m. Solution: To find the perimeter, simply add up the lengths of the sides: P = 4(9) = 36 yd. Perimeter of a Rectangle In general, if L is the length of a rectangle and w is the width of the rectangle, then the formula for the perimeter of a rectangle is P = 2L + 2w. Perimeter of a Square In general, if s is the length of the side of a square, the formula for the perimeter of a square is P = 4s. w L s Find the following: Ex. 5 A rectangular playground is 80 feet by 50 feet. If the fencing costs $14 per yard, how much will it cost to enclose the playground? Solution: First, find the perimeter of the playground: P = 2(80) + 2(50) = 160 + 100 = 260 feet Next, convert the 260 feet into the yards: 260 ft 1yd 260 260 ft = • = yd = 86.6666… yd 1 3 ft 3 Now, treat the $14 per yard as a conversion: 86.6666...yd $14 86.6666… yd = • ≈ $1213.33 yd 1 € € € € € 129 Objective #4: Find the sum of the angles of a polygon. Find the sum of the measures of the angles of the polygons below: Ex. 6 Ex. 7 Solution: We can split any quadrilateral into two triangles: Solution: We can split any pentagon into three triangles: The sum of the angles The sum of the angles in each triangle is 180˚, so in each triangle is 180˚, so the sum for quadrilateral is the sum for pentagon is 2•180˚ = 360˚. 3•180˚ = 540˚. In general, the number of triangles needed is two less than the number of sides of the polygon. This leads to following formula: Sum of the Angles of a Polygon The sum of the measures of the angles of a polygon with n sides is Sum = (n – 2)180˚ Solve the following: Ex. 8a Find the sum of the measures of the angles of a heptagon. Ex. 8b Find the measure of each angle of a regular heptagon. Solution: a) Since a heptagon has 7 sides, then the sum of the angles of a heptagon is (7 – 2)180˚ = (5)180˚ = 900˚ b) All the angles will have the same measure in a regular heptagon. Thus, the measure of each angle will be: 4 900˚ ÷ 7 = 128 ˚ 7 Ex. 9 Find the measure of ∠B in the diagram to the right. Assume the€figure is a regular polygon. B 130 Solution: The sum of the measures of the angles of hexagon is (6 – 2)180˚ = 720˚. Since the hexagon is a regular polygon, each angle has a measure of 720˚ ÷ 6 = 120˚. ∠B is the supplement of the angle on the bottom right of the hexagon. Thus, m∠B = 180˚ – 120˚ = 60˚. Objective #5: Understanding the angle requirements for tessellations. A tessellation of a flat surface is the tiling of the surface using one or more geometric shapes, called tiles, in a regular pattern leaving no overlaps and no gaps. This is often used to tile a floor or wall with mosaics. Some examples of tessellations using regular polygons are illustrated below: Squares Equilateral Triangles Regular Octagons & Squares Regular Hexagons Regular Hexagons & Equilateral Triangles Equilateral Triangles & Squares At each vertex in a tessellation, the same number and types of polygons must be used and the sum of the measures of the angles must equal to 360˚. Otherwise, it will not be a tessellation. 131 Show that each figure is a tessellation: Ex. 10 Solution: The sum of the measures of the angles of a hexagon is 4(180˚) = 720˚. Thus, each angle is 720˚ ÷ 6 = 120˚. Since each vertex is surrounded by three regular hexagons and the sum of the measures of the angles at each vertex is 120˚ + 120˚ + 120˚ = 360˚, then the figure is a tessellation. Ex. 11 Solution: The sum of the measures of the angles of an octagon is 6(180˚) = 1080˚. Thus, each angle is 1080˚ ÷ 8 = 135˚. Each vertex is surrounded by two regular octagons and one square. Also, the sum of the measures of the angles at each vertex is 135˚ + 135˚ + 90˚ = 360˚, so the figure is a tessellation. Ex. 12 Explain why a tessellation cannot be formed using regular decagons only. Solution: The sum of the measure of the angles of decagon is 8(180˚) = 1440˚. Each angle of a regular decagon is 1440˚ ÷ 10 = 144˚. Two regular decagons fill in 2•144˚ = 288˚, leaving a gap of 360˚ – 288˚ = 72˚. There can be no gaps with tessellations so a tessellation cannot be formed with using only regular decagons.