Chapter 3 Geometry：3章几何.doc

Chapter 3. Geometry English for mathematics Page 60 Chapter 3: Geometry Unit 1: INTRUCTION TO GEOMETRY _ 1. VOCABULARY AND GRAMMAR REVIEW: 1.1Vocabulary: -Abstract algebra (n): i s tru tng (i s i cng) -Algebraic geometry (n): Hnh hc i s -Ambient (a): xung quanh -Analytic geometry (n): Hnh hc gii tch -Area (n): din tch -Astronomy (n): Thin vn hc -Axiomatic (a): Thuc v tin -Analytic geometry (n): Hnh hc gii tch -Cartesian coordinates (n): H ta cc -Coastline (n): b bin -Commutative algebra (n): i s giao hon. -Complex analysis (n): gii tch phc -Concurrent (a): cng thi gian, ng quy -Differential geometry (n): Hnh hc vi phn -Euclidean geometry (n): Hnh hc clit -Exemplify (v): minh ha bng th d - Field (n): trng -Figure (n): hnh -Framework (n): sn, khun kh -Geometer (n): nh hnh hc -Geometry (n): Hnh hc -Homogeneous (a): thun nht -Intrigue (a): hp dn -Isometric (a): cng kch thc -Intrinsic (a): bn cht -Manifold (n): a tp -Mariner (n): thy th -Millennia (n): Thin nin k -Non-Euclidean geometry (n): Hinh hc phi clit -Projective geometry (n): Hnh hc x nh -Provenance (n): ngun gc -Scope (n): phm vi, tm, kin thc -String (n): chui -Transformations (n): s thay i, s bin i -Tuple (n): b -Volume (n): th tch Chapter 3. Geometry English for mathematics Page 61 1.2 Grammar review: - Relative clause - Preposition 1.3 Exercises: 1.3.1 Choose the best word to fill in the blanks. WhoWhoseByToOf Geometry is a part _(1)_ mathematics which has contributed to the development of ideas in other subject areas. For instance, geometry contributed _(2)_ the calculations of the early European mariners _(3)_ mapped the coastline _(4)_ Australia. It has also _(5)_ used extensively _(6)_ artists throughout the ages, including M.C. Escher, _(7)_intriguing designs are very popular. 1.3.2 Fill in the blanks with the suitable form of the words in the brackets: Algebraic geometry _(1)_(be)is a branch of mathematics which _(2)_(combine) techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It _(3)_(occupy) a central place in modern mathematics and _(4)_(have) multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it _(5)_(become) at least as important to understand the totality of solutions of a system of equations, as to find some solution; this leads into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique. Descartess idea of coordinates is central to algebraic geometry, but it has undergone a series of _(6)_(remark) transformations beginning in the early 19th century. Before then, the coordinates were assumed to be tuples of real numbers, but this changed when first complex numbers, and then elements of an arbitrary field became acceptable. Homogeneous coordinates of projective geometry offered an extension of the notion of coordinate system in a different direction, and enriched the scope of algebraic geometry. Much of the development of algebraic geometry in the 20th century _(7)_(occur) within an abstract algebraic framework, with increasing emphasis being placed on intrinsic properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels _(8)_(develop) in topology and complex geometry. 2. READING 2.1 GEOMETRY Geometry is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatmentEuclidean geometryset a standard for many centuries to follow. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia. A mathematician who works in the field of geometry is called a geometer. There are some common branches of Chapter 3. Geometry English for mathematics Page 62 geometry are: Analytic geometry, algebraic geometry, differential geometry, Euclidean geometry and projective geometry etc. The introduction of coordinates by Ren Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane, curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of calculus in the 17th century. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry. In Euclids time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation, and the question arose which geometrical space best fits physical space. With the rise of formal mathematics in the 20th century, also space (and point, line, plane) lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which space, point etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavor. While the visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance. Comprehension check: Answer the following questions: 1. According to the text, what is geometry? 2. Who put geometry into axiomatic form? 3. In 17th century, what marked a new stage in geometry? 4. How can we call a mathematician working in the field of geometry? 5. When was non-Euclidean geometry discovered? 6. What is the characteristic of modern geometry? 7. What are some branches of geometry? 3. SPEAKING LISTENING WRITING - DISCUSSION 3.1. Discussion: 1. Do you like studying geometry? Why and Why not? 2. In order to be good at geometry, what should we do? 3. What are some types of geometry that you have studied? What are their characteristics? 3.2 Writing: Write a short paragraph about a famous geometer you know. 3.3 Listening: Chapter 3. Geometry English for mathematics Page 63 Listen to the tape and fill in the blanks: _(1)_ is a part of mathematics which has contributed to the _(2)_ of ideas in other subject areas. For instances, geometry contributed to the _(3)_ of the early European mariners who mapped the coastline of Australia. It has also been used extensively by _(4)_ throughout ages. The world we live in is _(5)_dimensional. We often call these dimensions _(6)_; width and depth (or thickness). However, we have difficulty representing our three-dimensional world in _(7)_. This is because we have only _(8)_ dimensions, namely length and width, available to us when we are drawing. To overcome this _(9)_we try to _(10)_ the eye by drawing lines at angles to simulate the effect of depth. The isometric grid paper is very useful when we are drawing in this mode. 4. TRANSLATION: 4.1. Translate into Vietnamese: 4.1.1 Translate each of the following sentences into Vietnamese: a) The world we live in is three dimensional. We often call these dimensions length, width and depth (or thickness). b) The Swiss mathematician Leonhard Euler (pronounced “Oiler”) devised a formula that connects the number of edges (E), the number of faces (F) and the number of vertices in any polyhedron. Eulers rule states that: F + V 2 = E. c) In Euclids time there was no clear distinction between physical space and geometrical space. d) Geometry is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. e) Modern geometry has multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity. 4.1.2 Translate the following text into Vietnamese: Analytic geometry, branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic. Its most common application is in the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions. For example, the linear equation ax+by+c=0 represents a straight line in the xy-plane, and the linear equation ax+by+cz+d=0 represents a plane in space, where a, b, c, and d are constant numbers (coefficients). In this way a geometric problem can be translated into an algebraic problem and the methods of algebra brought to bear on its solution. Conversely, the solution of a problem in algebra, such as finding the roots of an equation or system of equations, can be estimated or sometimes given exactly by geometric means, e.g., plotting curves and surfaces and determining points of intersection. The methods of analytic geometry have been generalized to four or more dimensions and have been combined with other branches of geometry. Analytic geometry was introduced by Ren Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late 17th cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry. Chapter 3. Geometry English for mathematics Page 64 4.2. Translate into English: Vic hc ton cn phi suy ngh v lp lun. Sinh vin hiu r mt ch khng ch qua vic c v hc, m cn bng cch chng minh nh l v gii quyt vn . V th cc vn l mt phn quan trng trong ging dy, v chng gip sinh vin tranh lun, lp lun v hon chnh hn kin thc ca c nhn ca h. Puzzles: Cannibals ambush a safari in the jungle and capture three men. The cannibals give the men a single chance to escape uneaten. The captives are lined up in order of height, and are tied to stakes. The man in the rear can see the backs of his two friends, the man in the middle can see the back the man in front, and the man in front cannot see anyone. The cannibals show the men five hats. Three of the hats are black and two of the hats are white. Blindfolds are then placed over each mans eyes and a hat is placed on each mans head. The two hats left over are hidden. The blindfolds are then removed and it is said to the men that if one of them can guess what color hat he is wearing they can all leave unharmed. The man in the rear who can see both of his friends hats but not his own says, “I dont know“. The middle man who can see the hat of the man in front, but not his own says, “I dont know“. The front man who cannot see ANYBODYS hat says “I know!“ How did he know the color of his hat and what color was it? Just for fun: I had an amusing experience last year. After I had left a small village in the south of France, I drove on to the next town. On the way, a young man waved to me. I stopped and he asked me for a lift. As soon as he had got into the car, I said good morning to him in French and he replied in the same language. Apart from a few words, I do not know any French at all. Neither of us spoke during the journey. I had nearly reached the town, when the young man suddenly said, very slowly, “Do you speak English?” As I soon learnt, he was English himself. Unit 2: LINES ANGLES POLYGONS Chapter 3. Geometry English for mathematics Page 65 _ 1. VOCABULARY AND GRAMMAR REVIEW: 1.1 Vocabulary: -Acute angle (n): gc nhn -Acute triangle (n): tam gic nhn -Adjacent (a): k -Altitude (n): cao -Angle (n): gc -Cube (n): hnh lp phng -Complementary angle (n): gc b -Concave (a): lm -Congruent (adj): ng dng -Congruent (adj): ng dng -Convex (a): li -Coincident (a): trng khp -Circle (n): ng trn -Decagon (n): hnh 10 cnh -Dimension (n): chiu -Diagonal (n): ng cho -Equilateral triangle (n): tam gic u -Heptagon (n): hnh by cnh -Hexagon (n): hnh su cnh -Hypotenuse (n): cnh huyn ca tam gic vung -Intersect (v): giao nhau, ct nhau -Isosceles triangle (n): tam gic cn -Leg (n): cnh bn ca tam gic vung, cnh gc vung. -Line (n): ng thng -Line segment (n): on thng -Nonagon (n): hnh chin cnh -Obtuse angle (n): gc t -Octagon (n): hnh chn cnh -Plane (n): mt phng -Parallel (adj): song song -Parallelogram (n): hnh bnh hnh -Pentagon (n): hnh nm cnh -Perpendicular line (n): ng vung gc -Polygon (n): a gic -Proportional (a): t l -Proportional (n): s hng ca t l thc -Quadrilateral (n): t gic -Ray (n): tia -Reflex (a): phn x, to nh. -Rhombus (n): hnh thoi Chapter 3. Geometry English for mathematics Page 66 -Right angles (n): tam gic vung -Right isosceles triangle (n): tam gic vung cn -Solid (n): hnh khi (ba chiu) -Scalene triangle (n): tam gic thng -Similar triangles (n): tam gic ng dng -Straight angle (n): gc bt -Supplementary angle (n): gc ph -Subset (n): tp con -Side (n): cnh -Transverse (n): ng nm ngang. -Trapezoid (n): hnh thang -Triangle (n): tam gic -Trisection (n): chia lm ba -Vertex (n): nh. (plural form: vertices) -Vertical angle (n): gc i nh 1.2 Grammar review: -Conditional sentence -Tenses -Comparison 1.3 Exercises: 1.3.1 Fill in the blank with the suitable forms of the word in the bracket: If two straight lines meet at a point, they form an angle. The point _(1)_(call) the vertex of the angle, and the lines _(2)_(call) the sides or rays of the angle. Angles _(3)_(be) usually measured in degrees. Two angles are adjacent if they have the same vertex and a common side, and one angle _(4)_(be) not inside the other. If two lines intersect at a point, they form four angles. The angles opposite each other are called vertical angle. Vertical angles are _(5)_(equality). Lines that are parallel extend in the same direction and are the same distance apart at every point, so as never to intersect. The symbol / signifies that lines are parallel. When parallel lines are hit by _(6)_(transverse), all of the acute angles _(7)_(form) are congruent to each other, all the obtuse angles formed are congruent to each other, and every acute angle is _(8)_(supplement) to every obtuse angle. Perpendicular lines intersect such that they form or right angles. 0 90 1.3.2 Choose the correct word to fill in the blanks: Complementary; Supplementary; Acute; Obtuse; right; vertical; straight; reflex Two angles are _(1)_ if their measures sum to . Two angles are _(2)_if 0 90 their measures sum to . An _(3) _ angle is an angle whose measure is less than . An 0 180 0 90 _(4)_ angle is an angle whose measure is greater than but less than . A _(5) 0 90 0 180 _angle is an angle whose measure is exactly . Two pair of _(6)_ angles are formed 0 90 when two line intersect. _(7)_ angles are congruent. _(8)_angle is an angle that is exactly, while _(9)_ angle is an angle that greater than . 0 180 0 180 Chapter 3. Geometry English for mathematics Page 67 1.3.3 Put the word in column A with the suitable definition in column B AB a) Similar triangles 1) has one right angle and two acute angles. b) Isosceles triangle2) is triangle which all its three sides are congruent, and the three angles are also congruent. c) Equilateral triangle3) are triangles with congruent angles and proportional sides. d) Right triangle4) has at least two congruent sides, and the angles opposite these sides are also congruent. e) Hypotenuse5) is the side of a right triangle which is not opposite the right angle. f) Leg6) is the side opposite the right angle, in the right triangle. g) Right isosceles triangle7) is triangle which all its angles are acute. h) Acute triangle8) has a right angle and also two equal angles. i) Scalene triangle9) is a triangle which has neither equal sides nor equal angles. 1.3.4 Put the verbs in brackets in the correct forms: 1. The problem of constructing a regular polygon of nine sides which _(1)_(to require) the trisection of a angle _(2)_(to be) the second source of the famous 0 60 problem. 2. A plane can _(3)_(to draw) through a straight line and a point not on that line. 3. If two planes have a common point, then they have common straight line that _(4)_ (to pass) through that point (the line of intersection of the two planes); otherwise the planes ar