Areas of Polygons

We all have some idea about the quantity called area, from everyday life. We will establish here more precisely the concept of area of geometric figures, and develop methods for its computation. Assume that the area of a geometric figure is a quantity, expressed by positive numbers, and is well-defined for every polygon. Further assume that the areas of figures possess the following

properties:

(1) Congruent figures have equal areas. Figures of equal area are sometimes called equivalent. Thus, according to this property of areas, congruent figures are equivalent. The converse can be false: equivalent figures are not always congruent.

(2) If a given figure is partitioned into several parts, then the number expressing the area of the whole figure is equal to the sum of the numbers expressing the areas of the parts. This property of areas is called additivity. It implies, that the area of any polygon is greater than the area of any other polygon enclosed by it.

(3) The square, whose side is a unit of length, is taken for the unit of area, i.e. the number expressing the area of such a square is set to 1. When the unit of length is taken to be, say, 1 meter (centimeter, foot, inch, etc.), the unit square of the corresponding size is said to have the area of 1 square meter (respectively square centimeter, square foot, square inch, etc.), which is abbreviated as 1m² (respectively cm², ft², in², etc.). Measuring areas is done not by direct counting of unit squares or their parts fitting into the measured figure, but indirectly, by means of measuring certain linear sizes of the figure. Let us agree to call one of the sides of a triangle or parallelogram the base of those figures, and a perpendicular dropped to this side from the vertex of the triangle, or from any point of the opposite side of the parallelogram, the altitude. In a rectangle, the side perpendicular to the base can be taken for the altitude. In a trapezoid, both parallel sides are called bases, and a common perpendicular between them, an altitude. The base and the altitude of a rectangle are called its dimensions.

Theorem. The area of a rectangle is the product of its dimensions.

This brief formulation should be understood in the following way: the number expressing the area of a rectangle in certain square units is equal to the product of the numbers expressing the length of the base and the altitude of the rectangle in the corresponding linear units. It should be pointed out that the lengths of the base and the altitude (measured by the same unit) are expressed by whole numbers.

Unit 4

“Mathematics is the queen of sciences”.

Carl Friedrich Gauss

Grammar: The Infinitive. Its forms and functions.

Таблица форм инфинитива

	Active	Passive
Indefinite	to write	to be written	Выражает действие одновременное с действием глагола сказуемого
Continuous	to be writing	-	Выражает одновременное длительное действие
Perfect	to have written	to have been written	Выражает действие предшествующее действию глагола сказуемого (переводиться прошедшим временем)
Perfect Continuous	to have been writing	-	Выражает предшествующее длительное действие

Таблица функций инфинитива

1. Subject	To learn two foreign languages simultaneously is difficult. It’s useless to discuss the question.	Изучать два иностранных языка одновременно трудно. Бесполезно обсуждать этот вопрос.
2. Part of Predicate	Her task was to pass the exams.   He began to read the book a week ago.	Её задача состояла в том, чтобы сдать экзамены. Он начал читать книгу неделю назад.
3. Object	He doesn’t like to be asked personal questions.	Ему не нравится, когда задают вопросы личного характера.
4. Attribute (инфинитив переводится с оттенком модальности или будущности)	The problem to be discussed is interesting.   He was the first to do the exercise. There are some important things to be considered at the lesson.	Проблема, которая будет обсуждена (нужно, предстоит обсудить) интересна. Он первым сделал это упражнение. Существует несколько важных моментов, которые нужно рассмотреть на уроке.
5. Adverbial Modifier	I’m studying English in order (so as) to get a better job. To understand the importance of this event you should know all the facts.	Я изучаю английский язык для того, чтобы устроиться на лучшую работу. Чтобы понять важность этого события, вы должны знать все факты.
6. Compound Modal Predicate	He must be working now. She must have translated the text. They must have been reading the text-book for an hour. He may (might) be at his studies now. She may (might) have done the exercise.	Он, вероятно, сейчас работает. Должно быть, она уже перевела текст. Они, вероятно, читают этот учебник уже в течение часа. Возможно, он на занятиях сейчас. Она, возможно, выполнила упражнение.
Verbs which are usually followed by the Infinitive	agree, refuse, promise, threaten, offer, attempt, manage, fail, decide, plan, arrange, hope, appear, seem, pretend, afford, forget, learn (how), dare, tend	We promised not to be late for our classes. – Мы пообещали не опаздывать на занятия. He hoped to solve the problem at once. - Он надеялся решить проблему сразу.