Chapter 3. Geometry

Chapter 3: GeometryUnit 1: INTRUCTION TO GEOMETRY___________________________________

1. VOCABULARY AND GRAMMAR REVIEW:1.1 Vocabulary:- Abstract algebra (n): Đại số trừu tượng (đại số đại cương)- Algebraic geometry (n): Hình học đại số - Ambient (a): xung quanh- Analytic geometry (n): Hình học giải tích- Area (n): diện tích- Astronomy (n): Thiên văn học- Axiomatic (a): Thuộc về tiên đề- Analytic geometry (n): Hình học giải tích- Cartesian coordinates (n): Hệ tọa độ Đê các- Coastline (n): bờ biển- Commutative algebra (n): Đại số giao hoán.- Complex analysis (n): giải tích phức- Concurrent (a): cùng thời gian, đồng quy- Differential geometry (n): Hình học vi phân- Euclidean geometry (n): Hình học Ơclit- Exemplify (v): minh họa bằng thí dụ- Field (n): trường- Figure (n): hình- Framework (n): sườn, khuôn khổ- Geometer (n): nhà hình học- Geometry (n): Hình học- Homogeneous (a): thuần nhất- Intrigue (a): hấp dẫn- Isometric (a): cùng kích thước- Intrinsic (a): bản chất- Manifold (n): đa tạp- Mariner (n): thủy thủ- Millennia (n): Thiên niên kỷ- Non-Euclidean geometry (n): Hinh học phi Ơclit- Projective geometry (n): Hình học xạ ảnh- Provenance (n): nguồn gốc- Scope (n): phạm vi, tầm, kiến thức- String (n): chuỗi- Transformations (n): sự thay đổi, sự biến đổi - Tuple (n): bộ

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- Volume (n): thể tích1.2 Grammar review:-Relative clause-Preposition1.3 Exercises:1.3.1 Choose the best word to fill in the blanks.

Who Whose By To Of

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.

Descartes's 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. READING2.1 GEOMETRYGeometry 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 treatment—Euclidean geometry—set 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

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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 geometry are: Analytic geometry, algebraic geometry, differential geometry, Euclidean geometry and projective geometry etc.

The introduction of coordinates by René 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 Euclid's 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 20 th 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 - DISCUSSION3.1. 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.

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3.3 Listening: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. Euler’s rule states that: F + V – 2 = E.

c) In Euclid's 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

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René 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.

4.2. Translate into English:Việc học toán cần phải suy nghĩ và lập luận. Sinh viên hiểu rõ một chủ đề không chỉ qua

việc đọc và học, mà còn bằng cách chứng minh định lý và giải quyết vấn đề. Vì thế các vấn đề là một phần quan trọng trong giảng dạy, vì chúng giúp sinh viên tranh luận, lập luận và hoàn chỉnh hơn kiến thức của cá nhân của họ.

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 man's eyes and a hat is placed on each man's 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 don't know". The middle man who can see the hat of the man in front, but not his own says, "I don't know". The front man who cannot see ANYBODY'S 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.

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Unit 2: LINES – ANGLES – POLYGONS___________________________________________1. VOCABULARY AND GRAMMAR REVIEW:1.1 Vocabulary:

- Acute angle (n): góc nhọn- Acute triangle (n): tam giác nhọn- Adjacent (a): kề- Altitude (n): độ cao- Angle (n): góc- Cube (n): hình lập phương- Complementary angle (n): góc bù- Concave (a): lõm- Congruent (adj): đồng dạng- Congruent (adj): đồng dạng- Convex (a): lồi- Coincident (a): trùng khớp- Circle (n): đường tròn- Decagon (n): hình 10 cạnh- Dimension (n): chiều- Diagonal (n): đường chéo- Equilateral triangle (n): tam giác đều- Heptagon (n): hình bảy cạnh- Hexagon (n): hình sáu cạnh- Hypotenuse (n): cạnh huyền của tam giác vuông- Intersect (v): giao nhau, cắt nhau- Isosceles triangle (n): tam giác cân- Leg (n): cạnh bên của tam giác vuông, cạnh góc vuông.- Line (n): đường thẳng- Line segment (n): đoạn thẳng- Nonagon (n): hình chin cạnh- Obtuse angle (n): góc tù- Octagon (n): hình chín cạnh- Plane (n): mặt phẳng- Parallel (adj): song song- Parallelogram (n): hình bình hành- Pentagon (n): hình năm cạnh- Perpendicular line (n): đường vuông góc- Polygon (n): đa giác- Proportional (a): tỷ lệ

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- Proportional (n): số hạng của tỷ lệ thức- Quadrilateral (n): tứ giác- Ray (n): tia- Reflex (a): phản xạ, tạo ảnh.- Rhombus (n): hình thoi- Right angles (n): tam giác vuông- Right isosceles triangle (n): tam giác vuông cân- Solid (n): hình khối (ba chiều)- Scalene triangle (n): tam giác thường- Similar triangles (n): tam giác đồng dạng- Straight angle (n): góc bẹt- Supplementary angle (n): góc phụ- Subset (n): tập con- Side (n): cạnh- Transverse (n): đường nằm ngang. - Trapezoid (n): hình thang- Triangle (n): tam giác- Trisection (n): chia làm ba- Vertex (n): đỉnh. (plural form: vertices)- Vertical angle (n): góc đối đỉnh

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.1.3.2 Choose the correct word to fill in the blanks:

Complementary; Supplementary; Acute; Obtuse; right; vertical; straight; reflex

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Two angles are ___(1)____ if their measures sum to . Two angles are _____(2)___if their measures sum to . An __(3) ___ angle is an angle whose measure is less than . An __(4)___ angle is an angle whose measure is greater than but less than . A ____(5) __angle is an angle whose measure is exactly . Two pair of _____(6)___ angles are formed 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 . 1.3.3 Put the word in column A with the suitable definition in column B

A B

a) Similar triangles 1) has one right angle and two acute angles.

b) Isosceles triangle 2) is triangle which all its three sides are congruent, and the three angles are also congruent.

c) Equilateral triangle 3) are triangles with congruent angles and proportional sides.

d) Right triangle 4) has at least two congruent sides, and the angles opposite these sides are also congruent.

e) Hypotenuse 5) is the side of a right triangle which is not opposite the right angle.

f) Leg 6) is the side opposite the right angle, in the right triangle.

g) Right isosceles triangle 7) is triangle which all its angles are acute.

h) Acute triangle 8) has a right angle and also two equal angles.

i) Scalene triangle 9) 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 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

are coincident.4. A figure _____(5)___( to understand) to be a set of elements arranged in a plane or in

space: points, straight lines, rays, line segment (and sometimes planes).5. A solid ____(6)____(to be) ordinarily understood to be a portion of space bounded by

some closed surface. A cube is a solid bounded by six square faces. 6. Geometry ____(7)___(to subdivide) into plane geometry and solid geometry. Plane

geometry treats of the properties of various figures (triangles, polygons, circles) lying a plane. Solid geometry treats of the properties of three-dimensional figures and solids.

7. If two pair of sides of the triangles are proportional and the __(8)___(to include) angles are equal, the triangles are similar.

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8. If three sides of one triangle ____(9)___(to be) proportional to three sides of the other, the triangles are similar.

9. If two angles of one triangle are equal to two angles of the other, the triangles are similar (the third angles of the triangles ____(10)____(to turn out) to be equal too.

10. Gauss also ____(11)___(to demonstrate) the possibility of constructing a regular 257-gon via straightedge and compass.

1.3.5 Put the words in the correct order to make the meaningful sentence:a) The / mathematics / has / developed / world / around / us / contains / many / physical /

objects / from /which / geometric /ideas.b) Geometry/ not / includes / the / shape / only / and / size / of / the / earth / and / all /

things / on / it, / the / study / of / relations / but / also / between / geometric /objects.c) Between/ on / the / line / two / points,/ there / is /point / another.d) A / polygon / is / a / closed / figure / in / a plane / that / is / composed / of / line /

segments / that / meet / only / at / their /endpoints.e) A / figure/called / that / is/ convex / figure / not / is / a / concave.f) A / rhombus / is / a / four-sided / shape / where / all / sides / have / equal /length.

2. READING:2.1 GEOMETRY:

Geometry is a very old subject. It probably began in Babylonia and Egypt. Men needed practical ways, as the knowledge of the Egyptians spread to Greece, the Greeks found the ideas about geometry very intriguing and mysterious. The Greeks began to ask “Why? Why is that true?” In 300 B.C all the known facts about Greek geometry were put into a logical sequence by Euclid. His book, called Elements, is one of the most famous books of mathematics. In recent years, men have improved on Euclid’s work. Today, geometry includes not only the shape and size of the earth and all things on it, but also the study of relations between geometric objects. The most fundamental idea in the study of geometry is the idea of a point and line.

The world around us contains many physical objects from which mathematics has developed geometric ideas. These objects can serve as models of the geometric figures. The edge of a ruler, or an edge of this page is a model of a line. We have agreed to use the word line to mean the straight line. Geometric line is the property these models of lines have in common; it has length but no thickness and no width; it is an idea. A particle of dust in the air or a dot on a piece of paper is a model of a point. A point is an idea about an exact location; it has no dimensions. We usually use letters of the alphabet to name geometric ideas. For example, we speak of the following models of a point as point A, point B and point C.

.A .B

We speak to the following as line AB or line BA. A line is understood to be a straight line. A line is assumed to extend indefinitely in both directions. There is one and only one line between two distinct points.

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A BThe arrows on the model above indicate that a line extends indefinitely in both directions.

Let us agree to use the symbol to name a line. means line AB. Can you locate a point C between A and B on the drawing of above? Could you locate another point between B and C? Could you continue this process indefinitely? Why? Because between two points on the line, there is another point. A line consists of a set of points. Therefore, a piece of the line is a subset of a line. There are many kinds of subsets of a line. The subset of shown above is called a line segment. The symbol for the line segment B is . Points A and B are the endpoints, as you may remember. A line segment is a set of points on the line between them. How do line segment differ from a line? Could you measure the length of a line? Of a line segment? A line segment has definite length but a line extends indefinitely in each of its directions.

Another important subset of a line is called a ray. That part of shown below is called ray MN. The symbol for ray MN is

A ray has indefinite length and only one endpoint. The endpoint of a ray is called its vertex. The vertex of is M. In the drawings above you see pictures of a line, a line segment and a ray – not the geometric ideas they represent.

A line segment is the part of a line between two points called endpoints. Comprehension check:

1. Are the statements True (T) or False (F)? Correct the false sentences.a. A point is an idea about any dot on a surface.b. A point does not have exact dimension and location.c. We can easily measure the length, the thickness and the width of a lined. A line is limited by two endpoints.e. A line segment is also a subset of a line.f. Although a ray has an endpoint, we cannot define its length.2. Answer the following questionsa. Where did the history of geometry begin?b. Who was considered the first starting geometry?c. What was the name of the mathematician who first assembled Greek geometry in a

logical sequence?d. How have mathematicians developed geometric ideas?e. Why can you locate a point C between A and B on the line ?f. How does a line segment differ from a line, a ray?2.2 POLYGONSA polygon is a closed figure in a plane that is composed of line segments that meet only at

their endpoints. The line segments are called sides of the polygon, and a point where two sides meet is called a vertex (plural vertices) of the polygon. A diagonal of a polygon is a line segment whose endpoints are nonadjacent vertices. The altitude from a vertex P to a side is the line segment with endpoint P that is perpendicular to the side.

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Polygons are classified by the number of angles or sides they have. A polygon with three angles is called a triangle; a four-side polygon is a quadrilateral; a polygon with five angles is a pentagon; a polygon with six angles is a hexagon. A seven-sided polygon is a heptagon, while an eight-sided polygon is an octagon. A nine-sided polygon is nonagon, finally a ten-sided polygon is decagon. The number of angles is always equal to the number of sides in a polygon, so a six-sided polygon is a hexagon, which has six angles. The term n-gon refers to a polygon with n sides. A figure is convex if every line segment drawn between any two points inside the figure lies entirely inside the figure. A figure that is not convex is called a concave figure. A polygon is said to be regular if all sides and all angles are equal.

A trapezoid (called a trapezium in the UK) has one pair of opposite sides parallel. It is called an isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown. A trapezoid is not a parallelogram because only one pair of sides is parallel. The parallel sides are called bases and the nonparallel sides are called legs. A rhombus is a four-sided shape where all sides have equal length. Also opposite sides are parallel and opposite angles are equal.

If the sides of a polygon are all equal in length, and if all the angles of a polygon are equal, the polygon is called a regular polygon. If the corresponding sides and the corresponding angles of two polygons are equal, the polygons are congruent. Congruent polygons have the same side and the same shape.

If all the corresponding angles of two polygons are equal and the lengths of the corresponding sides are proportional, the polygons are said to be similar. Similar polygons have the same shape but need not to be the same size.

The sum of all the angles of an n-gon is . For example, the sum of the angles in a hexagon is .

Comprehension checkAnswer the following question:1) What is a polygon? Give some special polygons you have known?2) How can we create a diagonal and a altitude of the arbitrary polygon?3) How are polygons classified? Give some examples?4) What is the relationship between the number of sides and angles in a polygon?5) What are the characteristics of congruent polygons? Similar polygons? 6) Are congruent polygons also similar polygons? Why or why not?7) What is the sum of all the angles of the 20-gon?8) What are the characteristics of an isosceles trapezoid?9) Is a rhombus also a parallelogram? Why or why not?10)Is a trapezoid also a parallelogram? Why or Why not?11)What is the regular triangle? Regular quadrilateral? 12)Find the sum of the angles of each of the following: hexagon, heptagon, octagon,

nonagon and decagon. 13)In your opinion what is the suitable heading for this text?

3. WRITING – LISTENING - SPEAKING – DISCUSSION 3.1 Writing: Based on the above reading texts, continue the following definition:

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1. A point is …………………………………………………………………..2. A line is ……………………………………………………………………3. A line segment is …………………………………………………………..4. A ray is …………………………………………………………………….5. A pentagon is ………………………………………………………………6. A quadrilateral is …………………………………………………………..7. A trapezoid is ………………………………………………………………..8. A rhombus is ………………………………………………………………../9. A concave polygon is …………………………………………………………10. Similar polygons are …………………………………………………………11. Equivalent polygons are ……………………………………………………….12. A regular polygon is …………………………………………………………….13. An isosceles trapezoid is ………………………………………………………..3.2 Discussion:

Read the following text:Hero of Alexandria (1st century) was a Greek mathematician who invented several

pneumatic toys, including one known as “Hero’s fountain”. Hero’s name is also sometimes written as Heron. His name is given to a mathematical formula which can be used to find the area of a triangle if its side lengths are known. However, the formula was probably first used from before Hero’s time.

Hero’s formula is , where A = area of the trianglea, b, and c are the side lengthss = half the perimeter

a) Give the main idea of the text.b) Use Hero’s formula to find out the area of a triangle ABC if AB = 6cm, AC = 5cm, and BC = 3cm.c) Beside Hero’s formula, give some more formulae used to find out the area of a triangle, and give examples.4. TRANSLATION:

4.1 Translate into Vietnamese4.1.1 Translate the following sentences into Vietnamese: a) A rhombus is a four-sided shape where all sides have equal length. Also opposite sides

are parallel and opposite angles are equal.b) Polygons are classified by the number of angles or sides they have.c) A diagonal of a polygon is a line segment whose endpoints are nonadjacent vertices.d) Similar polygons have the same shape but need not to be the same size. 4.1.2 Translate the following text into Vietnamese:

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A triangle is a three-sided figure where the measures of the three angles sum to . In a triangle, sides opposite congruent angles are also congruent. Sides are related to each other in the same way that the angles opposite them are related; the greatest angle is opposite the greatest side.

The measure of the exterior angle of a triangle is equal to the sum of the measures of its tow remote interior angles. The perimeter of a triangle is equal to the sum of its three sides.

The area of a triangle is equal to (base x height). Every triangle has 3 bases, since any of its

three sides can be considered base. The height of a triangle is the perpendicular distance from the base to the vertex opposite it. Regardless of which side you consider as the base, the area will always be the same. The three sides of a triangle are related such that the third side must be greater than the positive difference of the other two sides and less than their sum.

4.2 Translate into EnglishĐịnh lý Pythagor: Trong tam giác vuông, bình phương cạnh huyền bằng tổng bình

phương hai cạnh góc vuông. Tứ giác là đa giác có 4 cạnh, trong đó tổng của 4 góc là . Hình bình hành, hình chữ

nhật, hình vuông là các tứ giác. Hình bình hành là tứ giác có hai cặp cạnh đối song song và bằng nhau. Các cặp góc đối

diện của nó thì bằng nhau và các các cặp góc liền kề thì bù nhau. Đường chéo chia hình bình hành thành hai tam giác bằng nhau. Trong hình bình hành thì tổng các bình phương của hai đường chéo bằng tổng các bình phương của các cạnh.

Hình chữ nhật là hình bình hành với bốn góc vuông. Các cặp cạnh đối diện của chúng thì bằng nhau. Chu vi hình chữ nhật bằng 2 (l + w) với l là chiều dài và w là chiều rộng. Diện tích của hình chữ nhật bằng lw.

Hình vuông là hình chữ nhật với bốn cạnh bằng nhau. Các đường chéo của nó chia đôi hình vuông. Chu vi hình vuông bằng 4s, với s là cạnh của hình vuông, và diện tích của hình vuông bằng .5. PRACTICE EXERCISES:

5.1 The man with his shadow and the flagpole with its shadow can be regarded as the pairs of corresponding sides of two similar triangles. What is the height of the flagpole?

h

6 50 4

45.2 The angles in a triangle are in the ratio of 2:3:5. What is the number of degrees in the

smallest angle of the triangle.5.3 What is the perimeter of a regular pentagon whose sides are 6 inches long?5.4 Is ABCD a parallelogram if A = (3, 2); B = (1, -2); C = (-2, 1) and D = (1, 5).

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5.5 City A is 5 miles north of city B, and city C is 12 miles west of city B. How far is it from city A to city C?

5.6 The length of a rectangle is l and the width is w. If the width is increased by 2 units, by how many units will the perimeter be increased?

5.7 If 8x represents the perimeter of a rectangle and 2x + 3 represents its length, what is it width?

5.8 A triangle has a base b and an altitude a. A second triangle has a base twice the altitude of the first triangle, and an altitude twice the base of the first triangle. What is the area of the second triangle?

5.9 The distance from A to C in the square field ABCD is 50 feet. What is the area of the field ABCD in square feet?

5.10 ABJH, JDEF, ACEG are squares. If then what is the ratio of

5.11 ABCD is a square AE = 2, GC = 8, shaded area = 44. Area of FBEJ = ?

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Puzzles:George adores classical music. He always prefers Beethoven to Bartok and Mahler to

Mozart. He always prefers Haydn to Hindemith and Hindemith to Mozart. He always prefers Mahler to any composer whose name begin with B, except Beethoven, and he always chooses to listen to a composer he prefer. Which of the following cannot be true?

A. George prefers Mahler to BartokB. George prefers Beethoven to MahlerC. George prefers Bartok to MozartD. George prefers Mozart to BeethovenE. George prefers Mahler to Haydn

Just for fun:Teacher: If the size of an angle is , we call it a right angleStudent: Then teacher, should we call other angles wrong angles?

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Unit 3: CIRCLE – CONIC SECTIONS AND SPACE FINGURES_______________________________________________________1. VOCABULARY – GRAMMAR REVIEW:1.1 Vocabulary:Vocabulary

- Chord (n): dây cung- Axis of symmetry (n): trục đối xứng- Circle (n): đường tròn- Centre (n): tâm - Coincide (v): trùng nhau- Circumference (n): chu vi- Chord (n): dây cung- Central angle (n): góc ở tâm- Concentric (a): đồng tâm- Cone (n): hình nón- Cross-section (n): mặt cắt- Cube (n): hình khối- Cylinder (n): hình trụ- Diameter (n): đường kính- Magnitudes (n): độ lớn- Directrix (n): đường chuẩn- Ellipse (n): hình Êlip- Inscribed angle (n): góc nội tiếp- Inscribed circle (n): đường tròn nội tiếp- Fixed point (n): điểm cố định- Flat face (n): mặt phẳng- Foci (n): tiêu điểm- Hyperbola (n): hình Hyperbol- Nondegenerate (a): chưa rút gọn- Parabola (n): hình parabol- Polyhedron (n): khối đa diện - Prism (n): hình lăng trụ- Pyramid (n): hình chóp- Radius (sing)– radii (plural) (n): bán kính- Sector (n): hình quạt- Cross-product term (n): số hạng dạng tích- Secant (n): cát tuyến.- Segment (n): hình viên phân- Sphere (n): hình cầu

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- Semicircle (n): nửa đường tròn- Space figure (n): vật thể trong không gian- Transverse axis (n): đường kính lớn (elip).- Orthogonal (a): trực giao - Linear transformation (n): phép biến đổi tuyến tính- Tangent (n): đường tiếp tuyến- Tetrahedron (n): tứ diện- External contact (v): tiếp xúc ngoài- Volume (n): thể tích

1.2 Grammar review:- Relative clauses- Conditional sentences- Passive voices

1.3 Practice exercises:1.3.1 Choose the correct words to fill in the blanks

Depth width objects space cubes polyhedron bothNot is to surface area six

A space figure or three-dimensional figure is a figure that has _____(1)___in addition to _____(2)___ and height. Everyday ____(3)___ such as a tennis ball, a box, a bicycle, and a redwood tree are all examples of _____(4)___ figures. Some common simple space figures include ____(5)___, spheres, cylinders, prisms, cones, and pyramids. A space figure having all flat faces is called a____(6)____. A cube and a pyramid are ____(7)___ polyhedrons; a sphere, cylinder, and cone are ____(8)___. A cross-section of a space figure ____(9)___ the shape of a particular two-dimensional "slice" of a space figure.

Volume is a measure of how much space a space figure takes up. Volume is used ____(10)__ measure a space figure just as area is used to measure a plane figure. The volume of a cube is the cube of the length of one of its sides. The volume of a box is the product of its length, width, and height. The _____(11)___of a space figure is the total area of all the faces of the figure. A cube is a three-dimensional figure having ____(12)___ matching square sides. If L is the length of one of its sides, the volume of the cube is L3 = L × L × L. A cube has six square-shaped sides. The surface area of a cube is six times the area of one of these sides.1.3.2 Put the words in the bracket in the correct form to fill in the blank.

THE MUTUAL POSITIONS OF TWO CIRCLES- The centres of the circles coincide. These ___(1)___(call) concentric circles. If the

____(2)___(radius) of these circles are not equal, then one of the circles lies inside the other. If the radii are equal, the circles coincide.

- Let (O, R) and (O’, r) are two circles, where the centres are distinct. We join them with a straight line____(3)___(call) the line of centres of the given pair of circles. The mutual positions of the circles will depend solely on the relationship between the magnitude of the line segment d joining their centres and the magnitudes of the radii of the circles, R and r.

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a) If the distance between the centres is less than the ____(4)___(differ) between the radii: d < R – r, the ___(5)___(small) circle lies inside the larger one.

b) If the distance between the centres is equal to the difference between the radii: d = R – r, the smaller circle lies inside the larger one, but they both have a common point on the line of centres.

c) The distance between the centres exceeds the difference between the radii, but is less than their sum: R – r < d< R + r. Each of the circles lies ____(6)___(part) within the other and partly outside. The circles have two points of ___(7)___(intersect).

d) The distance between the centres is ___(8)___(equality) to the sum of the radii: d = R + r. Each of the circles lies outside the other but they have a common point on the line of centres (external contact).

e) The distance between the centres exceeds the sum of the radii: d > R + r. Each of the circles is ___(9)___(entire) separate from the other. The circles do not have any points in common.

1.3.3 Put the words in the correct order to make meaningful sentences:a) A / given /circle / is / the / set / of / all / points / are / the/ which / same / distance /

from / a / point. b) A / radius/ point / on / the / some / is / a / line / segment / drawn / from / the / centre /

of / the / circle / to / circle. c) The / circumference / the / of / a / is / the / distance / circle / around / circle. d) If / is / called / the / chord / has / the / special / property / that / it / through / passes / the

/ centre / of/ the / circle / it / a / diameter. e) A / central / angle / of / a / circle / is / an / angle / which / of / the / circle / has / its /

vertex / at / the/ centre. f) A / line / which / touches / the / circle / in / exactly / one / the / circle / place / is /

called / a / tangent/ to.2. READING:2.1 CIRCLE

A circle is the complete set of points a given distance from a fixed point called its center. A circle is denoted by a single letter, usually its center. Two circles with the same center are concentric. The distance from the center to any point on the circle is called a radius. All radii of a circle are equal. A secant is a straight line having two common points with the circle. A chord of a circle is a line segment whose endpoints are on the circle. The diameter is a chord that passes through the center of the circle. The diameter is twice the length of a radius and is the longest chord. Every diameter is an axis of symmetry of the circle, as it divides the circle into two equal semicircles. The circumference of a circle is equal to or , where r is its radius and d its diameter. The area of a circle is equal to . A tangent to a circle is a line that touches the circumference of the circle at only one point and a sector is a fractional part of a circle’s area. The point common to a circle and a tangent to the circle is called the point of tangency. The radius from the center to the point of tangency is perpendicular to the tangent. A straight line passing through a point on the circumference of a circle is tangent to the circle if and only if it is perpendicular to the radius drawn to that point.

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An angle whose vertex is a point on a circle and whose sides are chords of the circle is called an inscribed angle. An angle whose vertex is the center of a circle and whose sides are radii of the circle is called a central angle.

Comprehension checkAnswer the following questions:1) What is a circle? How are concentric circles?2) What is a chord of a circle? What is a sector?3) If a diameter of a circle is 20cm, what are the area and the circumference of that circle?4) What is an inscribed angle of a circle? How about central angle?5) What is the characteristic of a tangent to a circle?6) What is the difference between diameter and a chord of a circle?7) A circle with diameter d has area A. What is the area of a circle with diameter 2d?8) A circle is inscribed within a square whose side length is 5. What is the area of the

circle?3. WRITING – SPEAKING – LISTENING – DISCUSSION:3.1 Writing:3.1.1 Base on the above text continue the following definitions:

- Circle is ……………………………………………………………………..- Diameter of the circle is ……………………………………………………….- Concentric circles are …………………………………………………………- A tangent to the circle is …………………………………………………….- A radius of a circle is ………………………………………………………- A secant is …………………………………………………………………- A chord is………………………………………………………………….

3.1.2 Write a short paragraph (more than 400 words) about the definition and characteristics of the following space figures: a cube; a cylinder, a pyramid, a prism, a sphere. 3.2 Discussion:

a) Give the definition of conic figures. List some formulae of ellipse, parabola and hyperbola that you have ever learned and draw their shapes.

b) Give the formulae to find out the total surface area (TSA) of a cylinder, a sphere, a cone, a pyramid. Then give examples. 3.3 Listening:Listen to the tape and fill in the blanks:

The total ____(1)____ area (TSA) of a solid is the combined areas of all____(2)____ faces of the solid. For example, the TSA of a_____(3)____ block is made up 6 rectangles; the two ends, the two _____(4)____, the top and the _____(5)____. In most cases, when working with ____(6)_____, the TSA is found by calculating the separate ____(7)____ and then adding

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them _____(8)____. The solids that have ______(9)____ formulae for TSA cylinders, _____(10)___ and spheres.

However, a pyramid is ____(11)___ up of a base and ____(12)___ sides. The number of triangles depends _____(13)___ the number of sides of the ______(14)___ shape; for instance, a pyramid with a _____(15)____ base is made up of four triangles, whereas a pyramid with a _____(16)____ base is made up of five triangles and so on. The TSA is once again ____(17)____ by adding the areas of the shapes forming the ____(18)____. Moreover, a solid sometimes is a composite of several others. To ______(19)___ the TSA of this type of solid you must find the _____(20)____ of each section ____(21)___. 4. TRANSLATION:

4.1. Translate into Vietnamese:4.1.1 Translate each of the following sentences into Vietnamese1) The set of points which are equidistant from a fixed point and a fixed line is a parabola.

The fixed point is the focus of the parabola and the fixed line is its directrix.2) The set of points, the sum of whose distances from two fixed points is a constant, is an

ellipse. The fixed points are the foci of the ellipse and the constant is the length of its major diameter.

3) The set of points, the differences of whose distance from two fixed points is a constant, is a hyperbola. The fixed points are the foci of the hyperbola and the constant is the length of its transverse axis.

4) To indentify a nondegenerate conic section Whose graph is not in standard position, we proceed as follows:

a) If a cross-product term is present in the given equation, rotate the xy-coordinate axes by means of an orthogonal linear transformation so that in the resulting equation the xy-term no longer appear.

b) If an xy-term is not present in the given equation, but an term and an x-term or an term or an y-term appear, translate the xy-coordinate axes by completing the square so that the graph of the resulting equation will be in standard position with respect to the origin of the new coordinate system.

4.1.2 Translate the following text into Vietnamese:Coordinate GeometryIn coordinate geometry, every point in the plane is associated with an ordered pair of

numbers called coordinates. Two perpendicular lines are drawn, the horizontal line is called the x-axis, and the vertical line is called the y-axis. The point where the tow axes intersect is called the origin. Both of the axes are number lines with the origin corresponding to zero. Positive numbers on the x-axis are to the right of the origin, negative numbers to the left. Positive numbers on the y – axis are above the origin, negative numbers below the origin. The coordinate of a point P are (x, y) if P is located by moving x units along the x-axis from the origin and then moving y units up or down. The distance along the x-axis is always given first.

The numbers in parentheses are the coordinates of the point. Thus P = (3, 2) means that the coordinate of P are (3,2). The distance between the point with coordinate (x, y) and the point with coordinate (a, b) is .

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4.2 Translate into English:4.2.1 Translate each of the following sentences into English:1. Một đường thẳng vuông góc với một mặt phẳng nếu nó vuông góc với một đường thẳng

bất kỳ của mặt phẳng đó. 2. Một đường thẳng vuông góc với hai đường thẳng cắt nhau của một mặt phẳng thì vuông

góc với mặt phẳng đó.3. Góc của một đường thẳng và một mặt phẳng là góc giữa đường thẳng đó và hình chiếu

vuông góc của đường thẳng đó trên mặt phẳng.4. Hình chóp được tạo nên bởi một mặt đáy và các mặt bên là các hình tam giác. 4.2.2 Translate the following text into English:René Descartes là một trong những nhà bác học và nhà triết học lừng danh ở thời đại của

ông, ông được xem như là một trong những người sáng lập ra triết học hiện đại. Sau khi tốt nghiệp đại học ngành luật, ông tự nghiên cứu toán học. Năm 1649, ông nhận lời mời của nữ hoàng Thụy Điển Christina để làm gia sư riêng cho bà và mở viện khoa học ở đó. Tuy nhiên, ông không thực hiện được kế hoạch này vì ông mất do bệnh viêm phổi năm 1650.

Năm 1619, trong một giấc mơ ông nhận ra rằng phương pháp toán học là cách tốt nhất để tìm ra chân lý. Tuy nhiên, ông chỉ có tác phẩm toán học duy nhất là La Géometri. Trong quyển sách này, ông đưa ra ý tưởng cơ bản về giải toán hình học bằng phương pháp đại số. Để biểu diễn một đường cong bằng phương pháp đại số, ta phải chọn một đường thuận tiện để làm cơ sở, và trên đường đó, chọn một điểm cơ sở. Nếu y chỉ khoảng cách từ một điểm bất kỳ trên đường cong đến đường cơ sở và x thể hiện khoảng cách từ đường thẳng cơ sở đến điểm cơ sở, khi đó có phương trình thể hiện mối liên hệ giữa x và y thể hiện đường cong. Phương pháp sử dụng hệ trục tọa độ của Cartesian được trình bày ở trên được giới thiệu vào thế kỷ 17 bởi các nhà toán học tiếp tục công việc nghiên cứu của Descartes.5. EXERCISES:

5.1 If L is the length of a cylinder, and r is the radius of one of the bases of a cylinder, then what is the volume of the cylinder? The surface area?5.2 If r is the radius of a sphere, What is the formula of the volume V of the sphere? The

surface area of the sphere?5.3 If r is the radius of the circular base, and h is the height of the cone, then what is the

formula of the volume of the cone?5.4 A picture in art museum is 6 feet wide and 8 feet long. If its frame has a width of 6

inches, what is the ratio of the area of the frame to be the area of the picture?5.5 A man travels 4 miles north, 12 miles east and then 12 miles north. How far (to the

nearest mile) is he from the starting point?5.6 When the radius of a circle is doubled, the area is multiplied by?

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5. 7 O is the center of the circle at the right, XO is perpendicular to YO and the area of triangle XOY is 32. What is the area of circle O?

5.8 Indentify the graph of the given equation.a) b) c) d)e) f) 5.9 Rewrite the following equation in standard form.

6. READING FOR REFERENCES:The evolution of our present – day meanings of the terms “ellipse”, “hyperbola”, and

“parabola” may be understood by studying the discoveries of history’s great mathematicians. As with many other words now in use, the original application was different from the modern.

Pythagoras (c.540 B.C), or members of his society, first used these terms in connection with a method called the “application areas”. In the course of the solution (often a geometric solution of what is equivalent to a quadratic equation) one of three things happens: the base of the constructed figure either falls short of exceeds, or fits the length of a given segment. (Actually, additional restrictions were imposed on certain of the geometric figures involved). These three conditions were designated as ellipsis “defect”, hyperbola “excess” and parabola “a placing beside”. It should be noted that the Pythagoreans were not using these terms in reference to conic sections.

In the history of conic sections, Menaechmus (350 B.C) a pupil of Eudoxus, is credited with the first treatment of conic sections. Menaechmus was led to the discovery of the curves of conic sections by a consideration of sections of geometrical solids. Proclus in his summary reported that the three curves were discovered by Menaechmus; consequently, they were called the “Menaechmian triads”. It is thought that Menaechmus discovered the curves now known as the ellipse, parabola and hyperbola by cutting cones with planes perpendicular to an element and with the vertex angle of the cone being acute, right or obtuse respectively.

The fame of Apollonius (C.225 B.C) rests mainly on his extraordinary conic sections. This work was written in eight books, seven of which are presented. The work of Apollonius on conic sections differed from that of his predecessors in that he obtained all of the conic

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sections from one right double cone by varying the angle at which the intersecting plane cuts the element.

All of Apollonius’s work was presented in regular geometric form, without the aid of algebraic notation of present day analytical geometry. However, his work can be described more easily by using modern terminology and symbolism. If the cone is referred to a rectangular coordinate system in the usual manner with point A as the origin and with (x, y) as coordinates of any points P on the cone, the standard equation of the parabola (where p is the length of the latus rectum, i.e. the length of the chord that passes through a focus of the conic perpendicular to the principal axis) is immediately verified. Similarly, if the ellipse or

hyperbola is referred to a coordinate system with vertex origin, it can be shown that

or respectively. The three adjectives “hyperbolic”, “parabolic”, and “elliptic” are

encountered in many places in maths, including projective geometry and non Euclidean geometries. Often they are associated with the existence of exactly two, one or more of something of particular relevance. The relationships arises from the fact that the number of points in common with the so called line at infinity in the plane for the hyperbola, parabola and ellipse is two, one and zero respectively. Comprehension check:

A. Answer the following questions:a) What did the words “ellipse’, “hyperbola” and “parabola” mean at the outset?b) What did these terms designate?c) Who discovered conic sections?d) What led Menaechmus to discover conic section?e) What were the curves discovered by Menaechmus called?f) What did Menaechmus do to obtain the curves?g) Who supplied the terms “ellipse”, “parabola”, “hyperbola” referring to conic sections?B. Choose the suitable heading for the text?a) The discoveries of history’s great mathematicians.b) Menaechmus discovered three curves of conic sections by a consideration of sections of

geometrical solids.c) History of the terms “ellipse”, “hyperbola” and “parabola”

Puzzles: ElevatorA Man works on the 10th floor and always takes the elevator down to ground level at the

end of the day. Yet every morning he only takes the elevator to the 7th floor and walks up the stairs to the 10th floor, even when is in a hurry. Why?

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Unit 4: EUCLID AND EUCLIDEAN GEOMETRY___________________________________________

1. VOCABULARY AND GRAMMAR REVIEW

1.1 Vocabulary:- Algebraic topology (n): đại số tôpô- Analytic functions (n): giải tích hàm- Circularity (n): sự xoay vòng, hình vòng tròn- Dissertation (n): luận án tiến sĩ- Euclidean geometry (n): hình học Ơclit- Geodesic (a): thuộc về đo đạc- Papyrus (n): sách giấy cói- Plane geometry (n): hình học phẳng- Postulate (n): tiên đề- Proposition (n): định đề- Solid geometry (n): hình học lập thể

1.2 Grammar review:- It + be + adj + clause- Passive voice- Conditional sentences- Tenses- Modal verbs

1.3 Practice exercises:

1.3.1 Fill in the blanks with the correct form of the word in the bracket:

Euclidean Geometry __(1)__(be) the study of flat space. We can easily ____(2)___(illustration) these geometrical concepts by ____(3)___(draw) on a flat piece of paper or chalkboard. In flat space, we know such concepts as: The ___(4)___(short) distance between two points is one unique straight line; The sum of the angles in any triangle ____(5)___(equal) 180 degrees etc. In his text, Euclid stated his fifth postulate, the famous parallel postulate, in the following manner: If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles. Today, we know the parallel postulate as simply stating: “Through a point not on a line, there is no more than one line parallel to the line.” The concepts in Euclid's geometry remained unchallenged until the early 19th century.  At that time, other forms of geometry

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____(6)___(start) to emerge, called non-Euclidean geometries.  It was no longer assumed that Euclid's geometry could be used to describe all physical space. Non-Euclidean geometries are any forms of geometry that contain a postulate (axiom) which is equivalent to the negation of the Euclidean parallel postulate for example Riemannian Geometry and Hyperbolic geometry.

Riemannian Geometry (elliptic geometry or spherical geometry) is the study of curved surfaces. The study of Riemannian Geometry ____(7)__(have) a direct connection to our daily existence since we live on a curved surface called planet Earth.  In curved space, the shortest distance between any two points (called a geodesic) is not unique, for example, there ___(8)___(be) many geodesics between the north and south poles of the Earth (lines of longitude) that are not parallel since they intersect at the poles. The sum of the angles of any triangle is now always greater than 180° and there are no straight lines on the sphere.  As soon as you start to draw a straight line, it curves on the sphere.

Hyperbolic Geometry (also called saddle geometry or Lobachevskian geometry). Unlike Riemannian Geometry, it is more__(9)__(difficulty) to see practical applications of Hyperbolic Geometry.  Hyperbolic geometry does, however, have applications to certain areas of science such as the orbit prediction of objects within intense gradational fields, space travel and astronomy.  Einstein ___(10)___(state) that space is curved and his general theory of relativity uses hyperbolic geometry.1.3.2 Fill in the blanks with the suitable form of the word in the bracket

MATHEMATICAL LOGIC

In order to communicate ____(1)____(effective), we must agree on the precise meaning of the terms which we use. It’s necessary _____(2)____(define) all terms to be used. However, it is impossible to do this since to define a word we must use others words and thus circularity cannot _____(3)___(avoid). In mathematics, we choose certain terms as undefined and define the others by using these terms. Similarly, as we are unable to define all terms, we cannot ____(4)___(proof) the truth of all statements. Thus we must begin by assuming the truth of some statements without proof. Such statements which are assumed to be true without proof _____(5)___ (call) axioms. Sentences which are proved to be laws are called theorems. The work of a _____(6) (mathematics) consists of proving that certain sentences are (or are not ) theorems. To do this he must use only the axioms, undefined and define terms, theorems already proved, and some laws of logic which have been ____(7)____(careful) laid down.

2. READING:

2.1 EUCLID

Euclid is often referred to as the "Father of Geometry." It is probable that he attended Plato's Academy in Athens, received his mathematical training from students of Plato, and then came to Alexandria. Alexandria was then the largest city in the western world, and the center of both the papyrus industry and the book trade. Ptolemy had created the great library at Alexandria, which was known as the Museum, because it was considered a house of the muses for the arts and sciences. Many scholars worked and taught there, and that is where Euclid

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wrote The Elements. There is some evidence that Euclid also founded a school and taught pupils while he was in Alexandria.

The Elements is divided into thirteen books which cover plane geometry, arithmetic and number theory, irrational numbers, and solid geometry. Euclid organized the known geometrical ideas, starting with simple definitions, axioms, formed statements called theorems, and set forth methods for logical proofs. He began with accepted mathematical truths, axioms and postulates, and demonstrated logically 467 propositions in plane and solid geometry. One of the proofs was for the theorem of Pythagoras, proving that the equation is always true for every right triangle. The Elements was the most widely used textbook of all time, has appeared in more than 1,000 editions since printing was invented, was still found in classrooms until the twentieth century, and is thought to have sold more copies than any book other than the Bible.

Axioms are statements that are accepted as true. Euclid believed that we can't be sure of any axioms without proof, so he devised logical steps to prove them. Euclid divided his ten axioms, which he called "postulates," into two groups of five. The first five were "Common Notions," because they were common to all sciences:

1. Things which are equal to the same thing are also equal to one another.2. If equals are added to equals, the sums are equal.

3. If equals are subtracted from equals, the remainders are equal.

4. Things which coincide with one another are equal to one another.

5. The whole is greater than the part.

The remaining five postulates were related specifically to geometry: 6. You can draw a straight line between any two points.7. You can extend the line indefinitely.

8. You can draw a circle using any line segment as the radius and one end point as the center.

9. All right angles are equal.

10. Given a line and a point, you can draw only one line through the point that is parallel to the first line.

Comprehension check:

Answer the following questions:

1) Who is referred as the “Father of Geometry”?2) What did Euclid do when he was in Alexandria?3) What is The Elements about?

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4) Is the Elements widely used? Why or Why not?5) What is axiom? How many axioms did Euclid write in the Elements? What are they?6) In your opinions, what is the difference between axiom and theorem?7) What is the theorem of Pythagoras about?8) In your opinion, what is the axiom which shows the differences between Euclidean

geometry and non-Euclidean geometry?2.2 THE PYTHAGOREAN PROPERTY

The ancient Egyptians discovered that in stretching ropes of lengths 3 units, 4 units and 5 units as shown below, the angles formed by the shorter ropes is a right angle (Figure 1). The Greeks succeeded in finding other sets of three numbers which gave right triangles and were able to tell without drawing the triangles which ones should be right triangles, their method being as follow. If you look at the illustration you will see a triangle with a dashed interior (Figure 2).

5 4

3

Fig.1 Fig.2

Each side of t is used as the side of a square. Count the number of small triangular regions in the two smaller squares then compare with the number of triangular regions in the largest square. He Greek philosopher and mathematician Pythagoras noticed the relationship and was credited with the proof of this property. Each side of right triangle was used as a side of a square, the sum of the areas of the two smaller squares is the same as the area of the largest square.

Proof of the Pythagorean Theorem

We would like to show that the Pythagorean Property is true for all right angle triangles, there are several proofs of this property.

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c

b

a

a

b

ab

b

a2

b2 b

b

a b

aab

b

b

a b

a

b

aab/2

ab/22ab

2ab

Chapter 3. Geometry

Fig.3 Fig.4 Fig.5

Let us discuss one of them. Before giving the proof let us state Pythagorean Property in mathematical language. In the triangle (Figure 3), c represents the measure of the hypotenuse, and a and b represent the measures of the other two sides. If we construct squares on the three sides of the triangle, the area – measure will be and . Then the Pythagorean Property could be stated as follows: for the triangle above, construct two squares each side of which has a measure a + b as shown in figure 4 and figure 5.

Separate the first of the two squares into two squares and two rectangles as shown in figure 4. Its total area is the sum of the areas of the two squares and two rectangles.

In the second of two squares construct four right triangles as shown in figure 5. Are they congruent? Each of the four triangles being congruent to the original triangle, the hypotenuse has a measure c. It can be shown that PQRS is a square, and its area is . The total area of the second square is the sum of the areas of the four triangles and the square PQRS.

The two squares being congruent to begin with, their area measures are the same. Hence we may conclude the following:

By subtracting 2ab from both area measures we obtain which proves the Pythagorean Property for all right triangles.

Comprehension check:

1. Which sentences in the text answer these questions:a) Could the ancient Greeks tell the actual triangles without drawing? Which ones would

be triangles?b) Who noticed the relationship between the number of small triangular regions in the two

smaller squares and in the largest square?

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c) Is the Pythagorean Property true for all right triangles?d) What must one do to prove that for the triangle under consideration?e) What is the measure of the hypotenuse in which each of the four triangles is congruent

to the original triangle?2. Which sentences is the main idea of the text?a) The Pythagorean theorem is true for all right triangles and it could be stated as follows:

.b) The text shows that the Pythagorean Property is true for all right triangles.c) The Greek mathematician, Pythagoras contributed to maths history his famous theorem

which was proved to be true for all right triangles. 3. WRITING – SPEAKING – LISTENING – DISCUSSION:3.1 Writing: 3.1.1 Due to the above text, continue the definition:a) Euclidean Geometry___________________________________________________b) Riemannian Geometry ___________________________________________________c) Hyperbolic Geometry_____________________________________________________

3.1.2 Continue these statements to make a theorem about the mutual positions of a straight line and a plane or of two planes.

a) A straight line not in some plane and parallel to one of the straight lines in the plane is itself parallel to ____________

b) A straight line is parallel to a plane if and only if it doesn’t lie in that plane and is parallel to one of _________in that plane.

c) If two straight lines are parallel to a third line, they are parallel to _________.d) If two parallel planes are cut by a third plane, then their lines of intersection are _______e) Through a point outside a give plane, it is possible to draw only one plane parallel to the

given ______.f) If two planes are separately parallel to a third _______, they are all parallel.g) A straight line parallel to each one of two intersecting planes is parallel to the line of

their__________.3.1.3 Write a short paragraph to answer the following question:

What is the basic difference between Euclidean geometry and non-Euclidean geometry, such as Riemannian geometry and Hyperbolic geometry? What’re the characteristics of Riemannian geometry and Hyperbolic geometry?

3.2 Listening:Listen to the tape and fill in the blanksAnalytic geometry is a _____(1)___of mathematics that reduces the study of a geometric problem to the study of an algebraic_____(2)___. In this way, many geometric problems are

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____(3)____and their results are more easily interpreted. With the aid of analytic geometry it is also ____(4)____to give a geometric interpretation to many algebraic____(5)___. Réné Descartes,Who live from 1596 to 1650, was one of the first to ____(6)___the theory of algebra into the study of geometry. His ____(7)____were introduced to the word through his ____(8)___La Geometrie which appeared about 1637. Accordingly, analytic geometry is sometimes ____(9)___“Cartesian geometry”.4. TRANSLATION:4.1. Translate into Vietnamese:

1. The geometry of line – called one-dimensional geometry – includes the study of lines, rays and line segments.

2. A plane in can be determined by specifying a point in the plane and a vector perpendicular to the plane. This vector is called a normal to the plane.

3. Jules Henri Poincaré was born in Nancy, France, to a well-to-do family, many of whose members played key roles in the French government. As a youngster, he was clumsy and absent-minded but showed a great talent in mathematics. His doctoral dissertation dealt with the existence of solutions to differential equations. In pure mathematics he was one of the principal creators of algebraic topology and made numerous contributions to algebraic geometry, analytic functions and number theory. He was the first person to think of chaos in connection with his work in astronomy. 4.2. Translate into English:

1. Góc là giao của hai tia có chung một điểm gốc nhưng không cùng nằm trên một đường thẳng.

2. Vì một góc là giao của hai tập hợp các điểm, nên bản thân nó cũng là tập hợp các điểm. Khi ta nói “góc ABC” thì ta đang nhắc đến tập hợp các điểm – các điểm nằm trên hai tia.

3. Hai góc sau đây thường xuyên xuất hiện trong hình học nên được gọi dưới tên đặc biệt. Góc được gọi là góc vuông và góc là góc bẹt.

4. Có ba trường hợp cơ bản về vị trí tương đối của hai đường thẳng trong không gian. 1. Chúng nằm trên cùng một mặt phẳng và cắt nhau.2. Chúng nằm trên cùng một mặt phẳng và song song với nhau.3. Chúng chéo nhau, tức là chúng không cùng nằm trên một mặt phẳng.

Puzzles:Seven teenagers at Gateway Amusement Park – Carlos, Leona, Gregor, Ingrid Naomi,

Dave and Rick – are going to ride the new roller coaster, Dragon’s Breath. Two cars are available, but the teens have to split up according to the following conditions:

Carlos and Naomi are boyfriend and girlfriend and must be in the same car. Dave and Gregor are friends but Ingrid is Gregor’s girlfriend, so Dave cannot be in the

same car as Gregor unless Ingrid is also in that car. The roller coaster rules say that the maximum number of riders in each car is four. Leona is Gregor’s sister and Rick is Leona’s ex-boyfriend, so neither Leona nor Gregor can

ride in the same car as Rick. If Rick rides in the same car as Ingrid, which of the following must be true?

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A. Dave rides in the same car as LeonaB. Dave rides in the same car as CarlosC. Leona rides in the same car as GregorD. Naomi rides in the same car as Rick and IngridE. Dave rides in the same car as Naomi.

Just for fun:Student A: Which animals have four legs but can swim and fly?Student B: Two ducks!

1. Read and give the meaning of the following sentences and give examples:

a) The angle an arc forms at the center of a circle is twice the size of the angle formed on

the circumference.

b) The angle formed on the circumference from a diameter of a circle is always a right

angle.

c) All the angles at the circumference standing on the same arc are equal.

d) A cyclic quadrilateral is a quadrilateral in which all four vertices are on the

circumference of the one circle. The opposite angles in a cyclic quadrilateral add up to 1800.

(They are supplementary).

e) Any tangent drawn on a circle meets the radius of the circle at right angles.

f) The angle an arc forms at the centre of a circle is twice the size of the angle formed at the

circumference.

g) The angle formed on the circumference from a diameter of a circle is always a right

angle.

h) All the angles at the circumference standing on the same arc are equal.

i) The opposite angles in a cyclic quadrilateral add up to 1800.

2. Give the correct form of the word in the brackets.

a) Brahmagupta (6th century) ____(1)___ (to be) an Indian mathematician who _____(2)____ (to adapt) Hero’s formula for the area of a triangle into a formula for _____(3)___ (to find) the area of a cyclic quadrilateral. The _____(4)____ (to adapt) formula

is (to be): , where s = half the perimeter i.e. .

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ASSIGNMENT:

Chapter 3. Geometry

b) If we ______(5)____ (to wish) to find the length of any arc we use our _____(6)____ (to know) of ratios and fractions. The length of the arc _____(7)____ (to depend) on both the radius of the circle and the angle the arc forms at the center of the circle. If either, or both, of these increase so does the arc length. The total circumference of any circle _____(8)____(to give) by the formula and there are in a full circle. Consequently, the fraction of the circumference that the arc occupies ______(9)____ (to be) equivalent to the fraction of that the angle occupies, i.e. arc length = (angle size in degrees : 3600) x .

3. Translate the following text into Vietnamese:

When one shape is changed into another shape we say it has been transformed in to the new shape. If the change is brought about finding the image of the shape in a mirror we call it reflection. If it is brought about by turning the shape in relation to a particular point we call it rotation. If we just slide the shape across the page we call it translation. Reflection and rotation can both change the way the shape “looks”. Translation always retains the “look” of the shape.

Rotation can occur about one of the vertices of the shape or about a point completely separate from the shape. Rotations about a vertex are easier to visualize but we need to be able to handle both types. Translation is the easiest of the transformations to perform. The shape retains its orientation; it simply changes its position on the plane. All we need is for the size of the translation to be specified. This might be given as a number of units right or left, or a number of up or down, or as a combination of the two.

4 a) Choose the correct word to fill in the blanks.:

A three – dimensional _______(1)____ is a _____(2)___. It is made _____(3)___ edges, corners and faces. _____(4)_ these faces are polygons the solid is ____(5)____ a polyhedron. Remember, a polygon is a two-dimensional closed shape ______(6)___ straight sides.

A ______(7)____polyhedron is one with all faces the same shape. Such polyhedral _____(8)___ known as the Platonic solids (named after the _____(9)____ mathematician as philosopher Plato. Some _____(10)____ Platonic solids _____(11)___ tetrahedron; cube; octahedron; dodecahedron; icosahedron.

b) Translate the above text into Vietnamese

c) Write a short paragraph (more than 300 words) about the characteristics of octahedron; dodecahedron, and icosahedrons.

5) Reading:

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If Familiar Regular Figure Greek Are Called Up With Solid

Chapter 3. Geometry

If we mark any two points on a circle we divide the circle into two parts. Each of these parts is called an arc. If the arc makes up less than half the circumference it is called a minor arc. A minor arc is usually named by just two points, for example minor arc CD. The angle formed at the center by a minor arc is less than 1800. If the arc makes up more than a half of the circumference we call this a major arc. The angle formed at the center by a major arc is a reflex angle. If the central angle equals 1800 then the circle is divided into two equal arcs, each of which is called a semicircle.

If we draw an arc and join each of its endpoints to the center of the circle we form a sector. If the arc is less than half of the circumference of the circle we have a minor sector; if the arc is greater than half of the circumference we have a major sector.

Are these sentences true or false. Correct the false:1. When we divide the circle into two parts, the larger part of the circumference is a

major sector.2. The angle formed at the center by a major arc is greater than 1800.3. When we divide the circle into two arcs, we make a semicircle. 4. A minor sector is the arc less than half of the circumference of the circle.5. The minor arc is always less than half of the circumference.

6) Find out the mistake in each of the following sentences and correct it. 1. Axioms are statement that are accepted as true.

A B C D 2. The short distance between two points is one unique straight line.

A B C D 3. You can draw a circle used any line segment as the radius and one end point as the

center. A B C D

7) Choose one of these following topics, write a short text (more than 400 words), then give the main idea of the text in Vietnamese. Topic 1: The conics Topic 2: Equation of the line in 3-dimension space and in 2-dimension space Topic 4: Euclid and The Elements Topic 5: NonEuclidean geometry Topic 6: The sphere and solid____________________________________________________________________________

REVIEW:I. Choose the correct answer:1) Geometry is one of ________sciences concerning______, areas and volumes. A. old ….length B. older ….lengths C. oldest ….length D. the oldest …

lengths

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2) A…. who works in the field of geometry is called a …..A. mathematics ….geometric B. mathematician…. Goemeter C. Mathematician…

Geometer3) In Euclid’s time, there was no clear …. between physical space … geometrical space.A. distinct …or B. distinct …and C. distinction….in D. distinction…and4. The methods of analytic geometry ….. to four or more dimensions.A. has generalized B. have generalized C. has been generalized D. have been

generalized5. The world around us … many physical objects … mathematics has developed geometric

ideas.A. contains… which B. contain …in whichC. contains …from which D.is contained…that6. …. a ray has an endpoint, we … it length.A. Because… can’t define B. Although…. can defineC. However… can define D. Although… can’t define7. If two straight lines…. at a point, they … an angle.A. will meet … form B. met… would form C. meet … forms D. is met…are

formed8. We can’t easily measure…and …. of a lineA. the length, the thick… the width B. long, thick… wideC. the longest, the thickness, … the widest D. the length, the thickness… the width9. Two angles are … if they have the same …. and a common side, and one angle is not

inside the other. A. Adjacent … vertices B. Adjacent … vertex C. Vertical…vertices D. Vertical…

vertex10. When parallel lines… by…., all of acute angles formed are congruent to each other.A. are hit… transverse B. are hit… transversalC. hit.. transversal D.are extend… transverse11. A line segment has … length but a line extends … in each of its directions.A. definitely … indefinitely B. definite… indefiniteC. definite … indefinitely D. indefinite … definitely12. Polygons… by the number of angles or sides they have.A. classify B. are classifing C. are classified D. were classifiedII. Fill in the blanks with the correct form of the words in the bracket:Alexandre-Théophile Vandermonde (1735 – 1796) ____(1)___(to be) born and died in

Paris. His father was a _____(2)___(physics) who encouraged his son to pursue a career in music. Vandermonde ____(3)___(to follow) his father’s ____(4)___(to advise) and did not get interested in mathematics until he was 35 years old. His entire ____(5)____(mathematician) output consisted of four papers. He also ____(6)___(to publish) papers on chemistry and on the manufacture of steel. Although Vandermonde ___(7)___(to be) best known for his determinant, it ____(8)___(not appear) in any of his four papers. It ____(9)___(to believe) that someone mistakenly attributed this determinant to him. However, his fourth mathematical paper, Vandermonde made significant ____(10)___(contribute) to the theory of determinants.

III. Put the words in the correct order to make a meaningful sentence:

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1) If / straight /an / lines/ meet / at / two / a/ point / form / they /angle.2) The/ vertical/ opposite / angles / other / called / are / each / angle.3) Two / to / are / if / angles / their / sum/ complementary /measures / 900. 4) Two/ measures / supplementary/ to /angles / if / their / are / sum/ 1800. 5) An / whose / angle / is / than /an / angle / measure/ acute / is / less / 900.6) Polygons / angles / by / are / number / of / classified / or/ have / the / sides / they. 7) A / only / is / not / a / parallelogram/ trapezoid / is/ of / parallel/ because / one / pair /

sides . 8) Gauss / the / possibility / of / a / by /regular / constructing / 257-gon / ruler/ and /

demonstrated /compass.9) The / the / two/ point / called / axes / intersect / where / the /is / origin. 10) Mean / obtained / by / items /is / added / a / of / data / item / together/ and / dividing /

number / by / the / number / of / data / adding. 11) The / dx/ can / be / as / differential / of x/ referred / where / x / is / to / as / the/

considered / variable / of/ the / quantity / integration. IV. Translation:A. Translate into Vietnamese:1. Cylinder is a space figure having two congruent circular bases that are parallel. 2. A tetrahedron is a four-sided space figure whose each face is a triangle. 3. A cone is a space figure having a circular base and a single vertex. 4. The three adjectives “ hyperbolic” , “parabolic”, “elliptic” are encountered in many

places in maths, including projective geometry and non Euclidean geometries. B. Translate into English:1. Nếu hai đường thẳng cùng song song với đường thẳng thứ ba thì chúng song song với

nhau. 2. Toán học cung cấp công cụ và phương pháp nhằm giúp nhà khoa học dự đoán kết quả. 3. Xác suất của biến cố A bằng khả năng xảy ra của biến cố A chia cho tổng số các khả

năng có thể xảy ra. 4. Quy tắc Hospital là quy tắc để tìm giới hạn của tỷ số hai hàm số. V. Fill in the blanks with the suitable words.

Doctorate out in born career contributions theorydied inequality proof published mechanics physics

Viktor Yakovlevich Bunyakovsky was ____(1)___ in Bar, Ukraine. He received a ___(2)___ in Paris in 1825. He carried ___(3)___ additional studies in St. Peterburg and then had a long ____(4)__ there as a professor. He made important ____(5)___ in number ____(6)___ and also worked in geometry, applied ____(7)___ and hydrostatics. His ____(8)___ of the Cauchy – Schwarz ____(9)___ appeared in one of his monographs in 1859, 25 years before Schwarz ____(10)___ his proof. He ____(11)___ in St. Peterburg.

VI. Underline the words that doesn’t belong to its group1. Directrix; data; polygons; vertices; variables2. Concern; Concave; Complex; Convex; divisible3. Successive; Possible; Independent; Failure; Definite4. Trial; regularity; Binomial; Possibility; Variance5. Distribution; Approximation; Contribution; Progression.

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6. Variance; Variable; Variety; Vertices; Various. 7. Parallelogram; Square; Triangle;Rhombus; Rectangle

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