Chapter 3. Geometry
Chapter 3: Geometry
Unit 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
- Volume (n): thể tích
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Chapter 3. Geometry
1.2 Grammar review:
- Relative clause
- Preposition
1.3 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. 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 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
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
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Chapter 3. Geometry
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 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:
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Chapter 3. Geometry
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
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.
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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.
Unit 2: LINES – ANGLES – POLYGONS
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Chapter 3. Geometry
___________________________________________
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
- 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
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Chapter 3. Geometry
- 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 900 or right angles.
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 900 . Two angles are _____(2)___if
their measures sum to 1800 . An __(3) ___ angle is an angle whose measure is less than 900 . An
__(4)___ angle is an angle whose measure is greater than 900 but less than 1800 . A ____(5)
__angle is an angle whose measure is exactly 900 . Two pair of _____(6)___ angles are formed
when two line intersect. _(7)____ angles are congruent. ____(8)___angle is an angle that is
1800 exactly, while ___(9)___ angle is an angle that greater than 1800 .
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Chapter 3. Geometry
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 600 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.
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 257gon via straightedge and compass.
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Chapter 3. Geometry
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.
B
A 
The arrows on the model above indicate that a line extends indefinitely in both directions.
 to name a line. AB means line AB. Can you locate a
Let us agree to use the symbol 
point C between A and B on the drawing of AB 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 AB
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Chapter 3. Geometry
shown above is called a line segment. The symbol for the line segment B is AB . 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 MN shown below is called
ray MN. The symbol for ray MN is MN
A ray has indefinite length and only one endpoint. The endpoint of a ray is called its vertex.
The vertex of MN 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 line
d. 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 questions
a. 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 AB ?
f. How does a line segment differ from a line, a ray?
2.2 POLYGONS
A 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.
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 tensided 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.
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Chapter 3. Geometry
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 (n  2)1800 . For example, the sum of the angles in a
hexagon is (6  2)1800  7200 .
Comprehension check
Answer 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:
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 ………………………………………………………………../
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Chapter 3. Geometry
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 A  s(s  a)(s  b)(s  c) , where A = area of the triangle
a, b, and c are the side lengths
s = 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 Vietnamese
4.1.1 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:
A triangle is a three-sided figure where the measures of the three angles sum to 1800 . 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
1
(base x height). Every triangle has 3 bases, since any of its
2
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
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Chapter 3. Geometry
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 English
Định 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 3600 . 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 s 2 .
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
4
5.2 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).
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?
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Chapter 3. Geometry
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?
A
B
C
D
5.10 ABJH, JDEF, ACEG are squares. If
B
A
J
G
F
a
a
aE
a
a
a
a
a
a
a
a
aJ
a
a
a
Ga
a
a
a
a
a
a
f
BC
Area BCDJ
 3 then what is the ratio of
?
AB
Area HJFG
C
D
E
5.11 ABCD is a square AE = 2, GC = 8, shaded area = 44. Area of FBEJ = ?
A
H
D
B
F
C
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 Bartok
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Chapter 3. Geometry
B. George prefers Beethoven to Mahler
C. George prefers Bartok to Mozart
D. George prefers Mozart to Beethoven
E. George prefers Mahler to Haydn
Just for fun:
Teacher: If the size of an angle is 900 , we call it a right angle
Student: Then teacher, should we call other angles wrong angles?
Unit 3: CIRCLE – CONIC SECTIONS AND SPACE FINGURES
_______________________________________________________
1. VOCABULARY – GRAMMAR REVIEW:
1.1 Vocabulary:
Vocabulary
- Chord (n): dây cung
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Chapter 3. Geometry
- 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
- 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:
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Chapter 3. Geometry
- 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 both
Not
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.
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).
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Chapter 3. Geometry
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 2 r or  d , where r is its
radius and d its diameter. The area of a circle is equal to  r 2 . 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.
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 check
Answer 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?
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Chapter 3. Geometry
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
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 Vietnamese
1) 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.
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Chapter 3. Geometry
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 x 2 term and an x-term or an
y 2 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 Geometry
In 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 ( x  a) 2  ( y  b) 2 .
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:
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Chapter 3. Geometry
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?
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?
Y
X
O
5.8 Indentify the graph of the given equation.
a) 4 x 2  25 y 2  100  0
b) 9 y 2  4 x 2  36
c) x 2  4 y  0
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Chapter 3. Geometry
d) y 2  0
e) x 2  9 y 2  9  0
f) x 2  y 2  0
5.9 Rewrite the following equation in standard form.
x 2  4 y 2  6 x  16 y  23  0
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
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 y 2  px (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
x2 y 2
x2 y 2


1

 1 respectively. The three adjectives “hyperbolic”, “parabolic”, and
or
a 2 b2
a 2 b2
“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
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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: Elevator
A 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?
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
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- 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
____(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
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Chapter 3. Geometry
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
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
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Chapter 3. Geometry
axioms, which he called "postulates," into two groups of five. The first five were "Common
Notions," because they were common to all sciences:
1.
2.
3.
4.
5.
Things which are equal to the same thing are also equal to one another.
If equals are added to equals, the sums are equal.
If equals are subtracted from equals, the remainders are equal.
Things which coincide with one another are equal to one another.
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:
Who is referred as the “Father of Geometry”?
What did Euclid do when he was in Alexandria?
What is The Elements about?
Is the Elements widely used? Why or Why not?
What is axiom? How many axioms did Euclid write in the Elements? What are they?
In your opinions, what is the difference between axiom and theorem?
What is the theorem of Pythagoras about?
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).
1)
2)
3)
4)
5)
6)
7)
8)
5
4
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Chapter 3. Geometry
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.
a
a
a2
b
ab
b
a
a
ab
2
ab/2
a
b
ab
b
c
b
b2
a
b
b
ab
2
b
Fig.3
ab/2
b
Fig.4
a
Fig.5
b
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 a 2 , b 2 and c 2 . Then the Pythagorean Property
could be stated as follows: c 2  a 2  b 2 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.
A  a 2  2ab  b 2
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Chapter 3. Geometry
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 c 2 . The total area of the
second square is the sum of the areas of the four triangles and the square PQRS.
1
A  c 2  4( ab)
2
The two squares being congruent to begin with, their area measures are the same. Hence
we may conclude the following:
1
a 2  2ab  b 2  c 2  4( ab)
2
2
2
2
(a  b )  2ab  c  2ab
By subtracting 2ab from both area measures we obtain a 2  b 2  c 2 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?
c) Is the Pythagorean Property true for all right triangles?
d) What must one do to prove that c 2  a 2  b 2 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:
2
c  a 2  b2 .
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_____________________________________________________
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Chapter 3. 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 blanks
Analytic 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
____(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 3 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.
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Chapter 3. Geometry
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 900 được gọi là góc vuông và góc 1800 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?
A. Dave rides in the same car as Leona
B. Dave rides in the same car as Carlos
C. Leona rides in the same car as Gregor
D. Naomi rides in the same car as Rick and Ingrid
E. 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!
ASSIGNMENT:
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.
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Chapter 3. Geometry
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): A  (s  a)(s  b)(s  c)(s  d ) , where s = half the perimeter i.e. s 
abcd
.
2
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 C  2 r and there are 3600 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 2 r .
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
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Chapter 3. Geometry
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.:
If Familiar Regular
Figure
Greek
Are Called
Up
With
Solid
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:
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.
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Chapter 3. Geometry
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
2) A…. who works in the field of geometry is called a …..
A. mathematics ….geometric B. mathematician…. Goemeter C.
Mathematician…Geometer
3) In Euclid’s time, there was no clear …. between physical space … geometrical space.
A. distinct …or B. distinct …and
C. distinction….in
D. distinction…and
4. The methods of analytic geometry ….. to four or more dimensions.
A. has generalized
B. have generalized C. has been generalized D. have been
generalized
5. The world around us … many physical objects … mathematics has developed geometric
ideas.
A. contains… which
B. contain …in which
C. contains …from which
D.is contained…that
6. …. a ray has an endpoint, we … it length.
A. Because… can’t define
B. Although…. can define
C. However… can define
D. Although… can’t define
7. 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
formed
8. We can’t easily measure…and …. of a line
A. the length, the thick… the width
B. long, thick… wide
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Chapter 3. Geometry
C. the longest, the thickness, … the widest
D. the length, the thickness… the width
9. 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…vertex
10. When parallel lines… by…., all of acute angles formed are congruent to each other.
A. are hit… transverse
B. are hit… transversal
C. hit.. transversal
D.are extend… transverse
11. A line segment has … length but a line extends … in each of its directions.
A. definitely … indefinitely
B. definite… indefinite
C. definite … indefinitely
D. indefinite … definitely
12. Polygons… by the number of angles or sides they have.
A. classify
B. are classifing
C. are classified
D. were classified
II. 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:
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.
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Chapter 3. Geometry
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
theory
died
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 group
1. Directrix; data; polygons; vertices; variables
2. Concern; Concave; Complex; Convex; divisible
3. Successive; Possible; Independent; Failure; Definite
4. Trial; regularity; Binomial; Possibility; Variance
5. Distribution; Approximation; Contribution; Progression.
6. Variance; Variable; Variety; Vertices; Various.
7. Parallelogram; Square; Triangle;Rhombus; Rectangle
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