<Outline> Mathematical Training Program for Laotian Teachers

Transcription

1 <Outline> Mathematical Training Program for Laotian Teachers Euclidean Geometry Analytic Geometry Trigonometry Dr Wattana Toutip Department of Mathematics Faculty of Science Khon Kaen University February 2003 ** Details should be added during the discussing time! **

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3 1. Euclidean Geometry 1.1 Introduction In the ancient Greek, the word Geometry means earth (geo) measure (metria) which began studying by Euclid (300 BC.). However, in the present this word may be changed because its development is extent to various modern geometries. In this program we concentrate only on Euclidean geometry which is the basis of other Non- Euclidean geometries. We discuss the basic concept and some of former theorems and mention some new theorems leading the modern synthetic geometry. 1.2 Angle and Measurement Degrees Radians Problem 1.1 Why does the right angle be 90? Problem 1.2 Danger of construction in Euclidean geometry? 3

4 Problem 1.3 Why the sum of the angles of a triangle is more than 360 Problem 1.4 Can we trisect any angle using ruler and compass? 1.3 Axioms (1) Things that are equal to the same thing are also equal to one another. (2) If equals are added to equals, then the whole are equal. (3) If equals are subtracted from equals, then the remainders are equal. (4) Things that coincide with one another are equal to one another. (5) The whole is greater than the part. 1.4 The fifth postulate of Euclid (1) A straight line can be drawn from any point to any point (2) A finite straight line can be produced continuously in a straight line (3) A circle may be described with any point as center and any distance as radius (4) All right angles are equal to one another (5) If a transversal falls on two lines in such a way that the interior angle on one side of the transversal are less than two right angles, then the lines meet on that side on which the angles are less than two right angles. 4

5 1.5 Parallel line Problem 1.5 Why does the sum of internal angles of any triangle be 180? Problem 1.6 Are there any triangle whose the sum of internal angles is greater than 180? Problem 1.7 Are there any other geometries? Topology Projective Euclidean Elliptic Hyperbolic Spherical 5

6 1.6 Congruence of Triangles Problem 1.8 Danger of using ASA theorem? 1.7 Triangles and Similarity Problem 1.9 How to divide straight line into equal parts? Problem 1.10 Given a and b. How to construct straight lines with length a b, a b, ab and a b? Problem 1.11 Why does ( a)( b) ab? 6

7 1.8 Area and definition 1.9 Area of Polygon Problem 1.12 How to construct a square with the same area of a rectangle using only Euclidean equipment(ruler and compass)? 1.10 Area of Circle Problem 1.13 What is? Problem 1.14 Why does the area of a circle with radius r be 2 r? 7

8 1.11 Relations between Euclidean geometry and algebra ( a b) a 2ab b Problem 1.15 How to prove without words using Euclidean geometry? (1) ( a b) a 2ab b (2) 2 2 a b a b a b ( )( ) 1.12 Heron s Formula : If a, b and c are the opposite side of angles A, B and C a b c respectively and s then 2 Area s( s a)( s b)( s c) Problem 1.15 How to prove Heron s formula? 8

9 1.13 Pythagoras s Theorem: For a right triangle, the area of a square on the opposite side of the right angle equals the sum of areas of squares on the sides of the right angles. Problem 1.16 How to prove Pythagorean Theorem? (1) (2) (3) 9

10 Problem 1.17 What is an extension of Pythagorean Theorem? Problem 1.18 What is an important property of Pythagorean triple? Problem 1.19 Some theorems leading to modern synthetic geometry? 1.14 Menelaus s Theorem: Three points, one on each side of a triangle (extended if necessary) are collinear if and only if the product of the ratios of division of the sides by the points is 1 provided that internal ratios are considered positive and external ratios are considered negative. 10

11 Problem 1.20 How to prove Menelaus s Thorem? 1.15 Ceva s Theorem: Three lines that join three points, one on each side of a triangle, to the opposite vertices are concurrent if and only if the product of the ratios of division of the sides is 1. Problem 1.21 How to prove Ceva s Thorem? 11

12 Problem 1.22 Are there any applications of Ceva s Thorem? Problem 1.23 What is Analytic Geometry? In the 17 th century, French mathematicians Pierre de Fermat ( ) and Rene Descartes ( ) began using algebraic representations of figures. They realized that by assigning to each point in the plan and ordered pair of real numbers, algebraic techniques could be employed in the study of Euclidean geometry. This study of figures in terms of their algebraic representations by equations is known as analytic geometry. NOTE: 12

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14 2. Trigonometry 2.1 Introduction Trigonometry derived from the Greek words trigonon for triangle and metria for measurement. Initially, trigonometry involved the study of the relationships between the sides and angles of triangles. Today, the trigonometric functions come into play not only on when considering triangles but in many other areas of mathematics and its applications. In this work, we discuss the basic definition and some laws with their applications. 2.2 Definition Problem 2.1 Why does cos( ) cos? Problem 2.2 Why does sin( ) sin? 14

15 2.3 Trigonometry in Triangle 2.4 Law of Sine sin A sin B sin B a b c Problem 2.3 How to prove Law of Sine? Problem 2.4 How to apply Law of Sine? 15

16 2.5 Law of Cosine a b c 2bc cos A b a c 2ac cos B c a b 2ab cos C Problem 2.5 How to prove Law of Cosine? Problem 2.6 How to apply Law of Cosine? 16

17 Problem 2.7 How to prove Heron s Formula? 2.6 Some Identities of Trigonometric property (1) sin cos(90 ) 2 2 (2) cos sin 1 (3) cos( A B) cos Acos B sin Asin B (4) sin( A B) sin Acos B cos Asin B (5) sin 2A 2sin Acos A 2 2 (6) cos 2A cos A sin A Problem 2.8 How to prove some trigonometric identities without words? (1) sin cos(90 ) 90 (2) sin( ) sin cos cos sin a y b a = + y y b 17

18 (3) cos( ) cos cos sin sin (4) sin( ) sin cos cos sin (5) cos( ) cos cos sin sin 18

19 (6) sin 2A 2sin Acos A (7) cos 2A 1 2sin 2 A NOTE: 19

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