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Transcript
Mth 97 Fall 2013 Chapters 1 and 2 Problem Solving (Chapter 1) Polya’s 4 Steps: 1) Understand the problem; 2) Devise a plan; 3) Carry out the plan; 4) Look Back Problem Solving Strategies 1. Draw a Picture (See pages 5 and 6 of your text) – Work problem 2 on page 10. 2. Guess and Test (pages 6 – 7) – Work problem 17 on page 12 using toothpicks. 3. Use a Variable (pages 8 – 9) – Work the following problem. The perimeter of a picture is 26 inches. Its length is 3 inches greater than its width. What are the dimensions of the picture? (Hint: Define a variable, then write and solve an equation.) Section 2.1 – Undefined terms, postulates, segments and angles Axiomatic System: Undefined terms → Definitions → Postulates → Theorems Undefined terms: Can be described but cannot be given precise definitions Point – suggested a Line – suggested by a Space: The set of ________ points Plane – suggested by a Geometric Figure: Any collection of _____________ Collinear Points: Points that line on the same ____________. Non-collinear Points: Points that are ________ all on the same line. Coplanar Points: Points that lie in the same ________________. Non-coplanar Points: Points that are_______ contained in the same plane. Postulates: Statements that we assume to be ___________which state relationships between defined and undefined terms. 1 Mth 97 Fall 2013 Chapters 1 and 2 Theorems: Results deduced from undefined terms, definitions or postulates and can be _____________ Relationships Among Points, Lines, and Planes Postulate 2.1 – Every line contains at least two distinct points. Postulate 2.2 – Two points are contained in one and only one line. Postulate 2.3 – If two points are in a plane the line containing these points is also in the plane. Postulate 2.4 – Three non-collinear points are contained in one and only one plane and every plane contains at least three non-collinear points. Postulate 2.5 – In space, there exist at least four points that are not all coplanar. Line Segment: determined by its endpoints & includes all the points __________ these points on a line Postulate 2.6 (The Ruler Postulate) – Every line can be made into an exact copy of the real number line using 1–1 correspondence. A B C D E F G H I J -4 -3 -2 -1 0 1 2 3 4 5 The distance between two points is the ___________________ difference between their coordinates. AE = BF = DH = Congruent segments are segments that have the same______________________. AB DE because AB = DE = 1. CF CF CF because CF = ______ = ______ = _______ = ______ If the coordinates of A, B, and C are respectively 3 5 , 2 5 , and 5 , find each segment length. AB = BC = AC = A ray is a portion of a line that has one endpoint and extends indefinitely in _________ direction and is named by the endpoint followed by a point on the ray written below a ray symbol, A B C D 2 Mth 97 Fall 2013 Chapters 1 and 2 Angles and Their Measure Angle: Formed by two line segments or rays meeting at a common endpoint. Vertex ______ Sides _________ and _________ Named with an angle symbol and its vertex _____________ or an angle symbol and three points (side-vertex-other side) _____________ or ____________ Postulate 2.7 (The Protractor Postulate) If we place one ray of an angle at 0° on a protractor and we place the vertex at the midpoint of the bottom edge (or center crosshairs), then there is a 1-1 correspondence between all other rays that can serve as the second side of the angle and the real numbers between 0° and 180°. This number is called the “measure” of the angle. Measure the angle above and then do number 1 on today’s ICA (In-class Assignment). Types of Angles Acute– has a measure Right – has a measure of 90° between 0° and 90° Straight – has a measure of 180° Obtuse – has a measure between 90° and 180° Reflex – has a measure between 180° and 360° Congruent angles have the same ________________________ If the measure of DEF is 30° and DEF GHI , then the measure of GHI is ____________ Complementary – two angles whose sum is 90° Supplementary – two angles whose sum is 180° Do number 2 on today’s ICA. Adjacent angles – have a common vertex and side but no common interior points. 3 Mth 97 Given Find Fall 2013 Chapters 1 and 2 mAGB 75 mCGE 90 mCGD 50 mFGC 175 C B mDGE mFGE mAGF mBGC A E F In the drawing above name pairs of: Complementary angles D G Supplementary angles Adjacent angles Dimensional Analysis is used to make unit conversions. Review of Fractions 4 5 5 6 4 5 5 6 We use these two ideas along with unit fractions (fractions equal to 1) to do dimensional analysis. 12 in = 1 ft 1 hr = 60 min If you know how two units of measure are related, you can write a unit fraction. Instead of figuring out whether you want to multiply or divide by the relationship between two units of measure, we will always multiply by a unit fraction that allows us to CANCEL LABELS to achieve the desired unit of measure. More precise measures of angles include fractions of degrees or minutes and seconds, where 60 minutes = 1 degree (60’ = 1o) and 60 seconds = I minute (60” = 1’). Use dimensional analysis to express 32°15’40” in degrees as a decimal rounded to thousandths. Use dimensional analysis to express 15.12° in degrees and minutes. Use dimensional analysis to express 75.3° in degrees minutes and seconds. Complete the rest of ICA 1. 4 Mth 97 Fall 2013 Chapters 1 and 2 One radian is the measure of the central angle of a circle that cuts off a portion of the distance around the circle that is the same length as the radius of the circle. The radian measure of a full circle is 2π, so 2π rads = 360° or for half a rotation, π rads = 180°. Use dimensional analysis to find the radian measure for the following angles. unit frations 15° 120° Problem Solving Strategies (cont.) 4. Look for a Pattern (See page 16) Discuss problems 11a and 11b on page 25. 5. Use Make a Table (See pages 18-19) and inductive reasoning to solve problem 3 on page 24. 6. Solve a Simpler Problem (See page 21) Use this strategy to solve problems 17 on page 26. A 2.2 Polygons (2 dimensional shapes that lie in a plane) Triangles are determined by three _____________________________ points. ∆ABC is formed by __________ __________ and ___________ B C We can classify triangles by their sides and angles. Definition No two sides are Type Acute Definition All 3 acute angles Isosceles At least 2 sides are Right One right angle Equilateral All 3 sides are Obtuse One obtuse angle Equiangular All 3 angles are Angles Sides Types Scalene Sketch the following triangles: Isosceles Right ∆ Scalene Obtuse ∆ Scalene Acute ∆ Equilateral ∆ Obtuse Isosceles ∆ 5 Mth 97 Fall 2013 Chapters 1 and 2 Simple Closed Curves A simple closed curve is a figure that lies in a plane and can be traced so that the starting and ending points are the same and no part of the curve is crossed or retraced. Closed curve Not a closed curve A circle is a simple closed curve that consists of the set of all points ______________________ from a given point, called the center of the circle. A circle is named by its center point. A Polygon is a simple closed curve composed of line ______________________. Polygons are named listing the vertices in order around a figure. Adjacent vertices are the endpoints of a __________ of a polygon. Two sides that share a _______________ are adjacent. Polygons Not a polygon Polygons are usually described by the number of sides they have. Type of Polygon Triangle Quadrilateral Pentagon Hexagon Number of Sides Type of Polygon Octagon Decagon n-gon Number of Sides Parallel Lines lie in the same ______________ and do not intersect. Quadrilaterals are named by the lengths of their sides, the measure of their angles, or some other attribute such as parallelism. Quads Square Def. Rectangle Rhombus Parallelogram Trapezoid All sides are All angles are All sides and all right angles. are . angles are right angles. Isosceles Trapezoid Opposite sides Exactly one A trapezoid are parallel. pair of whose parallel non-parallel sides. sides are . Kite Two pairs of adjacent sides where the pairs have no side in common. Pix 6 Mth 97 Fall 2013 Chapters 1 and 2 Do problems 1 to 7 of ICA 2. Types of Symmetry in Polygons Reflection Symmetry: A figure has reflection symmetry if there is a ______________(axis of symmetry) along which the figure may be folded so that one half of the figure matches exactly the other half. Draw and cut out each of the following, then fold them to find all the lines of symmetry. Tell how many lines of symmetry each polygon has and sketch them on the drawings below. isosceles triangle equilateral triangle square rectangle ____ lines of symmetry ____ lines of symmetry ____ lines of symmetry ____ lines of symmetry Rotational Symmetry: a figure has rotational symmetry if it can be rotated about a point ______than a full turn, so that the image is identical to the original figure. If the above polygons have rotational symmetry, tell what angle(s) each shape must be rotated to achieve rotational symmetry. isosceles triangle rotational symmetry? _______ If yes, degree(s) rotated? ____________ equilateral triangle _______ ____________ square _______ ____________ rectangle _______ ____________ 2.3 Angle measure in Polygons and Tessellations Theorem 2.1 – Angle Measure in a Triangle The sum of the measures of the angles in a triangle is _______ Draw and cut out any triangle, then follow directions on page 74 for problem 35a & 35b. 53° Find the missing angle measures. x° 34° x= d° f° 65° a= d= b= e= c° a° 124° eq uat ion her c = e. f = e° b° 7 Mth 97 Fall 2013 Chapters 1 and 2 Do the rest of ICA 2 A E B The angles in a polygon are called ______________ angles. Find the sum of the measures of the vertex angles of a pentagon. Hint: Draw all possible diagonals from a single vertex count the number of triangles formed. D E Theorem 2.2 – Angle Measure in a Polygon The sum of the measures of the vertex angles in a polygon with n sides is (n – 2) 180°. A Regular Polygon is a polygon in which all the sides are congruent and all the angles are congruent. Which of the above polygons are regular? How could we find the measure of a vertex angle in a regular pentagon? Theorem 2.3 – Vertex Angle Measure in a Regular Polygon The measure of the vertex angle of a regular n-gon is Find the measure of the vertex angle of a regular hexagon. n 2 180 . n A vertex angle of a regular polygon measures 157.5°. How many sides does the polygon have? Use equations to solve problem 11 on page 72. a) b) c) d) 8 Mth 97 Fall 2013 Chapters 1 and 2 Tessellations A tessellation is a tile-like pattern formed by repeating _____________to fill a plane without gaps or overlaps. The shapes are polygonal regions, formed by each polygon together with the portion of the plane that it encloses. A regular tessellation (p. 70) is composed of regular polygons that are all the same ________and ___________. Which tessellation below is regular? 2.4 Three Dimensional Shapes (Solids) A Polyhedron is a three-dimensional shape composed of polygonal regions, any two of which have at most a common side. The plural is polyhedra. It must be enclosed with no ______________. The polygonal regions are called ____________________ The common line segments are called _____________________ The point where the edges meet is called a __________________ Page 80, problem 2: a) b) c) Types of Polyhedra: Prisms, Pyramids, Regular Polyhedrons A Prism is a polyhedron with two opposite identical faces, called __________, which are identical polygonal regions in ________________ planes. The other faces are called _________________ faces. The height of the prism is the ________________________ distance between the bases. Prisms are named by the shape of their ________________. Right Prisms have lateral faces that are _____________________________ regions. Oblique Prisms have lateral faces that are _______ rectangular regions. Page 82, problem 10: a) b) A Pyramid is a polyhedron consisting of a polygonal region for its ___________ and triangular regions as _________________ faces, which all meet at a point called the ____________ of the pyramid. The height is the ___________________ distance between the apex and the base. Pyramids are named by the shape of their _____________. 9 Mth 97 Fall 2013 Chapters 1 and 2 A right regular pyramid’s base is a _______________polygon; its lateral faces are ___________________ triangles; and its slant height is the height of any of its ___________________. An oblique regular pyramid’s lateral faces are NOT isosceles ___________________. Page 83, problem 13: a) ______________________________________________________ (3 or 4 words) b) ______________________________________________________ A Regular Polyhedron is a polyhedron in which all the _________are identical, regular polygons, that is, all the faces are regular polygons of exactly the same size and shape. Only five exist. Polyhedron Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron Shape of the face Triangle Square Triangle Pentagon Triangle # of faces (F) 4 6 12 20 #of vertices (V) 4 6 20 12 # of edges (E) 12 12 30 30 Euler’s Formula, F + V = E + 2, holds true for all polyhedra, not just regular polyhedra. Verify Euler’s Formula for a) Octahedron b) Square based pyramid c) Pentagonal prism Page 82, problem 11: Do page 1 of ICA 3 Other Solids: Cylinders, Cones, and Spheres A circular cylinder is formed by two congruent circles in _______________ planes together with the surface formed by line segments joining corresponding points of the two circles. The two circular regions are the ___________________ In a right circular cylinder, the segments joining corresponding points of the two bases are _________________________ to the bases. In an oblique circular cylinder, the segments joining corresponding points of the two bases are _______ perpendicular to the bases. A circular cone is formed by a circular region, called the ___________, together with the surface formed when the apex (a point not in the base) is joined by line segments to every point on the circle. 10 Mth 97 Fall 2013 Chapters 1 and 2 If the apex of a circular cone lies on the line that is perpendicular to the center of the base, the cone is a _________________ circular cone. Otherwise, the cone is ___________________. The height of a cone is the _______________________ distance to the base. The slant height in right circular cones is the distance from the apex to a point on the edge of the _____________. A sphere is the set of all points in space that are a fixed distance from a given point, the_____________ of the sphere. The ____________ is a segment whose endpoints are the center and a point on the sphere. The ________________ is a segment that contains the center of the sphere and whose endpoints are on the sphere. 2.5 Dimensional Analysis (Use “Units of Measurement” – the last page of the textbook) Ex.1: Covert 9 feet to inches. Ex. 2: Convert 50 pounds to ounces Ex. 3: A water bottle contains 24 ounces of water. How many quarts does it hold? Ex. 4: Convert 21 m into inches. Ex. 5: π radians = 180°. Use this to convert radians Ex. 6: How many square inches in 10 to degrees. square feet? Ex. 7: Lynn walks for exercise every day. She averages 6km/hr. What is her speed in m/sec? Do the rest of ICA 3. 11