Ex. 19. Read the text and find the answers to the following questions.

1. What are the main properties of the areas of geometric figures?

2. What numbers is the area of a geometric figure expressed by?

3. How is measuring areas done?

4. What is called the dimensions of a rectangle?

Areas of Polygons

We all have some idea about the quantity called area, from everyday life. We will Ex. 19. Read the text and find the answers to the following questions. 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 Ex. 19. Read the text and find the answers to the following questions. 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 Ex. 19. Read the text and find the answers to the following questions. 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 Ex. 19. Read the text and find the answers to the following questions. 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 Ex. 19. Read the text and find the answers to the following questions. said to have the area of l squaremeter (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 Ex. 19. Read the text and find the answers to the following questions. 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 Ex. 19. Read the text and find the answers to the following questions. 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 Ex. 19. Read the text and find the answers to the following questions.. 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 Ex. 19. Read the text and find the answers to the following questions. 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

ИНФИНИТИВ ФОРМЫ И ФУНКЦИИ ИНФИНИТИВА

Формы инфинитива

Active Passive
Indefinite to Ex. 19. Read the text and find the answers to the following questions. write to be written Выражает действие одновременное с действием глагола сказуемого
Continuous to be writing - Выражает одновременное долгое действие
Perfect to have written to have been written Выражает действие предыдущее действию глагола сказуемого (переводиться прошедшим временем)
Perfect Continuous to have been writing - Выражает предыдущее долгое действие

Функции инфинитива

1. Подлежащее Ex. 19. Read the text and find the answers to the following questions. (subject) To learn two foreign languages simultaneously is difficult. It’s useless to discuss the question. Учить два зарубежных языка сразу тяжело. Никчемно дискуссировать этот вопрос.
2. Часть сказуемого (part of Predicate) Her task was to pass the exams. He began to read the book two weeks ago Ex. 19. Read the text and find the answers to the following questions.. Её задачка состояла в том, чтоб сдать экзамены. Он начал читать книжку две недели вспять.
3. Дополнение (оbject) He doesn’t like to be asked personal questions. Ему не нравится, когда задают вопросы личного нрава.
4. Определение (attribute) (инфинитив переводится с цветом модальности либо будущности) The problem to be discussed is Ex. 19. Read the text and find the answers to the following questions. interesting. He was the first to do the exercise. There are some important things to be considered at the lesson. Неувязка, которая будет оговорена (необходимо, предстоит обсудить) увлекательна. Он первым сделал это упражнение. Существует несколько принципиальных моментов, которые необходимо разглядеть на уроке.
5. Событие (Adverbial Modifier) I’m studying English in order Ex. 19. Read the text and find the answers to the following questions. (so as) to get a better job. To understand the importance of this event you should know all the facts. Я изучаю британский язык для того, чтоб устроиться на наилучшую работу. Чтоб осознать значимость этого действия, вы должны знать все факты.
6. Часть составного модального сказуемого (compound мodal рredicate Ex. 19. Read the text and find the answers to the following questions.) 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, возможно, на данный момент работает. Должно быть, она уже перевела текст. Они, возможно Ex. 19. Read the text and find the answers to the following questions., читают этот учебник уже в течение часа. Может быть, он на упражнениях на данный момент. Она, может быть, выполнила упражнение.
Глаголы, после которых употребляется инфинитив agree, refuse, promise, threaten, offer, attempt, manage, fail, decide, plan, arrange, hope, appear, seem, pretend, afford, forget, learn (how), dare, tend We promised Ex. 19. Read the text and find the answers to the following questions. not to be late for our classes. – Мы пообещали не опаздывать на занятия. He hoped to solve the problem at once. - Он возлагал надежды решить делему сходу.


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