Урок 10. Circumference of a Circle. Exercises

Exercises on "Circumference of a circle", continuation

Упражнение 5. Выпишите из текста предложения, содержащие страдательную конструкцию.

1. In traditional approaches to mathematics the circumference of a circle, has not always been clearly defined.

2. That is, sometimes the circle itself was called the circumference, and at other times, the measure of the distance around the circle was called the circumference.

3. This definition is symbolized by the formula с = 2PR or the formula с = PD.

4. By using the limit concept, the circumference of a circle may be defined as the limit of the perimeter of an inscribed regular polygon.

5. Then, by bisecting the central angles which are subtended by the sides of the square we can inscribe a regular octagon.

6. Clearly the sum of "n” sides of an inscribed regular n-gon can be made to approximate the circumference of the circle as closely as desired by choosing "n” sufficJiently large.

7. Thus, the circumference of a circle may be defined as the limit of the perimeter of an inscribed regular n-gon as "n” increases.

Упражнение 6. Выпишите из текста предложения, в которых инфинитив употребляется, как обстоятельство цели.

1. To illustrate this, we can first inscribe a square in a circle.

2. Clearly the sum of "n” sides of an inscribed regular n-gon can be made to approximate the circumference of the circle as closely as desired by choosing "n” sufficJiently large.

Упражнение 7. Сгруппируйте слова по частям речи и переведите их.

1)      Approximate, approximately, approximation, to approximate.

2)      distant, distance.

3)      precise, precisely, precision;

4)      introduce, introduction, introductory;

5)      to close, close, closure, to enclose;

6)      circle, circular;

7)      limit, limitless, limitation, unlimited

8)      regular, regularly, regularity, irregular.

Упражнение 8. Переведите предложения и определите, к какой части речи относятся подчеркнутые слова.

1. These calculations are only approximate. (adjective – прил.)

Эти вычисления лишь приблизительны.

2. He spent half an hour to solve a problem based on the Pythagorian theorem. (Participle II)

Он потратил полчаса, чтобы решить задачу на теорему Пифагора.

3. Your definition of a circle is not precise. (adjective)

Твое определение круга неточное.

4. Have you read the introduction to the "Elements” by Euclid? (noun – сущ.)

Вы читали введение на «Началам» Евклида?

5. What else should be added to arrive at the more precise definition of a circumference? (Participle II)

Что еще следует добавить, чтобы определение окружности сделать более точным?

6. In the figure above you see an inscribed square. (Participle II)

На рисунке выше вы видите вписанный квадрат.

Упражнение 9. Напишите план текста.

1. Circumference in traditional approaches to mathematics.
2. Circumference as the perimeter of the circle.
3. Circumference in the limit approach.

Упражнение 10. Скажите, правильны ли следующие утверждения.

1. The area of a rectangle is the product of the length of its base and the length of its altitude. (T)

2. We can inscribe as many polygons in a circle as we desire.

(T. We can inscribe only regular polygons, and we can inscribe as many regular polygons as we desire.)

3. A radius of a circle is twice as long as a diameter.

(F, just the opposite, the diameter is twice as long as a radius. )

4. Opposite angles of a trapezoid are congruent.

(F, opposite angles of a rectangle or a parallelogram are congruent.)

5. A diameter of a circle is a chord of the same circle. (T)

6. The formula for the perimeter of a regular polygon may be stated as pr².

(F, it is p=n*t, n – the number of sides, t – the length of a side.)

7. If the intersection of two lines in space is an empty set then the lines are parallel. (F, they may also be skew [skju:] lines – скрещивающиеся)

8. If three parts of one triangle are congruent to three part of another, the triangles are congruent. (Т)

Упражнение 11. Прочтите текст и скажите, о чем говорится в этом тексте.

The circumference of a circle and the number

What is the distance around a given circle?

Какова длина окружности?

There are two convenient practical ways of finding such distance.

Существуют два удобных практических способа измерения такой длины.

We may think of the circle as the rim of a wheel.

Можно представить круг как обод колеса.

First we mark a spot on the rim, and then we roll the wheel along a flat surface.

Сначала делаем отметку на ободе, вращаем колесо по плоской поверхности.

Finally we measure the distance between two consecutive places where the mark has touched the surface.

Наконец, измеряем расстояние между двумя точками, где отметка коснулась поверхности.

Or we could wrap a string once around the wheel, and then find the length of the part of the string used.

Или мы могли бы обмотать шнур вокруг колеса, и потом найти длину использованной части шнура.

Those practical measuring techniques will provide approximations that are suitable for many purposes.

Эти практические методы обеспечат приблизительные результаты, которые подойдут во многих случаях.

But for mathematical purposes we require a precise definition.

Но для математики нужно точное определение.

A circle is an abstract object; no practical measuring techniques can be used to find the length of a circle.

Круг – это абстрактное тело; практические методы измерения длины круга не могут быть использованы.

We defined the perimeter of a polygon as the sum of the lengths of its sides, which are segments.

Мы определили периметр многоугольника как сумму длин его сторон, которые являются отрезками.

(That is easy to prove if you remember that no line intersects a circle in more than two points

Это легко доказать, если вы вспомните, что линия может пересекать окружность только в двух точках).

In order to formulate a mathematical definition for the length of a circle we shall use a property of the real number system that we haven’t yet discussed.

Чтобы сформулировать математическое определение длины окружности, нужно использовать свойство действительных чисел, которые мы еще не рассматривали.

This property is sometimes taken as a postulate of the real number system.

Это свойство часто рассматривается как постулат в теории действительных чисел.

Упражнение 12. Ответьте на следующие вопросы по образцу.

1. The distance between two points is being measured. What is the student doing? He is measuring the distance between two points.

2. New material is being introduced. Listen! What is the teacher doing?

The teacher is introducing new material.

3. Be attentive. A precise definition of a circumference is being given. What is the teacher doing?

The teacher is giving a precise definition of a circumference.

4. Peter’s homework is done regularly. What does Peter do?

Peter does his homework regularly.

5. These calculations can be made only approximately. How can you make these calculations?

I can make these calculations only approximately.

6. Look at the blackboard. An octagon is being inscribed in a circle. What is the student doing?

The student is inscribing an octagon in a circle.

7. The area of a rectangle can be symbolized by the following formula. By what formula can you symbolize the area of a rectangle?

I can symbolize the area of a rectangle by the following formula.

8. This rule can be illustrated by the following example. By what example can you illustrate this rule?

I can illustrate this rule by the following example.

9. Look, this theorem is being explained by the teacher on the blackboard. What is the teacher doing?

The teacher is explaining this theorem on the blackboard.

Упражнение 13. Ответьте на следующие вопросы по содержанию текста.

1. Can we say that a diameter of a circle is a chord of the same circle?

Yes, we can say that a diameter of a circle is a chord of the same circle.

2. Is the area of a circle greater or smaller than the area of an inscribed polygon?

The area of a circle is greater than the area of an inscribed polygon.

3. Does the area of a polygon approach that of a circle as the number of the sides of the polygon increases?

Yes, the area of a polygon approaches that of a circle as the number of the sides of the polygon increases.

4. Is P a rational number or not?

P is an irrational number.

5. Are all radii of the same circle congruent? Why?

Yes, all radii of the same circle are congruent, because the distance between the center of the circle and its circumference is equal in all directions.

6. Can the number P be expressed exactly as a fraction or as a decimal? Why not?

No, we cannot express P exactly as a fraction or as a decimal because Р is irrational.

7. Which is more accurate approximation of P: 13, 14 or 22?

More accurate approximation of P is 14.

8. Can we say that a circle and the circumference of a circle are one and the same thing?

We cannot say that.

9. How shall we define the circumference here?

The circumference of a circle may be defined as the limit of the perimeter of an inscribed regular polygon.

10. By what formula is the perimeter of circle symbolized?

The perimeter of circle is symbolized by the formula C = 2Pr.

11. What is it necessary to introduce if we are going to arrive at a more precise definition of a circle?

If we are going to arrive at a more precise definition of a circle, it is necessary to introduce the concept of limits.

12.  In what way can we inscribe a regular octagon in a circle?

We can firs inscribe a square in a circle. Then, by bisecting the central angles which are subtended by the sides of the square we can inscribe a regular octagon.

13. Can we continue the process of bisecting the central angle indefinitely?

Yes, we can continue the process of bisecting the central angle indefinitely.