Exercises to circles, radii, diameters, chords, arcs (continuation)

Exercises to circles, radii, diameters, chords, arcs (continuation)

Упражнение 11. Переведите следующие предложения на русский язык.

1. A circle can be easily drawn with the help of a compass.

Окружность легко начертить с помощью циркуля.

2. These points are fixed.

Эти точки фиксированы.

3. The line segment AR joints point A of the circle with its centre.

Отрезок АR соединяет точку А окружности с центром окружности.

4. Yes, there is an indefinite number of points contained in the circumference.

Да, на окружности содержится бесчисленное количество точек.

5. The diameter of a circle passes through its centre.

Диаметр окружности проходит через ее центра.

6. They considered the matter carefully.

Они изучили этот вопрос тщательно.

7. The ratio has been slightly changed.

Отношение было немного изменено.

8. This number is designated by a certain letter.

Это число обозначается определенной буквой.

9. Yes, points A and B representing the opposite points of a circle are equidistant from the centre.

Да, точки А и В, обозначающие противоположные точки окружности, находятся на одинаковом расстоянии от ее центра.

10. A circumference encloses a part of a plane.

Окружность ограничивает часть плоскости.

Упражнение 12. Скажите, какой ответ правильный.

What is a circle?

1. If you hold the sharp end of a compass fixed on a sheet of paper and then turn the compass around, you will draw a circle.
2. We call a circle a set of points each of which is equidistant from the centre. (Т)

What is a radius?

1. In any circle you can draw any number of radii.
2. A line segment joining any point of the circle with the centre is called a radius. (Т)

What term do we usually use to mean the distance around the circle?

1. A circumference encloses part of a plane.
2. We usually use the term circumference to mean the distance around the circle. (Т)

What is a diameter?

1. The diameter of a circle passes through its centre.
2. A chord which passes through the centre of the circle is a diameter. (Т)

What is a chord?

1. A diameter is a particular case of a chord.
2. A line segment whose endpoints are points on the circle is a chord of a circle. (Т)

Упражнение 13. Составьте план текста.

1. Definition of a circle. How to draw it.
2. Definition of a radius.
3. Definition of a chord.
4. Definition of an arc.

Упражнение 14. Скажите, правильны ли следующие утверждения.

1. а) A sector equal to a fourth part or quarter of a circle is called a quadrant ['kwPdrqnt]. (T)

b) A straight line drawn from the centre to any point on the circumference is called a chord. (F, it is a radius, a chord joins two points of the circle.)

2. a) When a circle is cut in half each of the halves is called a quadrant. (F, it is a semicircle)

b) A pair of compasses (циркуль) is a convenient instrument for drawing a circle. (T)

3. a) The centre of an arc is the centre of a circle which contains the arc. (F)

b) A semicircle is the union of the ends of a diameter and all the points of the circle lying on one side of the diameter. (T)

4. a) A circle is the locus of points in a plane equidistant from a given point and a given line. (F, a circle is the locus of points in a plane equidistant from a given point.)

b) The measure of an inscribed angle (вписанный угол) is one half the measure of its intercepted (отсеченный) arc. (Т)

5. a) A segment of a circle is that portion of the circle bounded by two radii of the circle and one of the arcs which they intercept. (F, it is a sector of a circle; a segment of a circle is a plane figure bounded by a chord and one of the arcs into which the chord divides the circumference.)

b) A diameter drawn to a point or contact is perpendicular to the tangent (касательная) at that point. (Т)

6. a) The length of an arc described by the endpoints of the radii is proportional to the magnitude (величина) of the corresponding central angle. (T)

b) The length of the circumference constitutes about 37 the length of the radius. (F)

Упражнение 15. Прочтите текст без словаря. Скажите, о чем говорится в этом тексте (на русском языке)

Equal circles are circles whose radii are equal.

Равные окружности имеют равные радиусы.

Concentric circles are circles which have the same centre.

Концентрические окружности – это окружности с одним и тем же центром.

A segment of a circle is a plane figure bounded by a chord and one of the arcs into which the chord divides the circumference.

Сегмент окружности – это часть плоскости, ограниченная хордой и одной из дуг, на которые хорда разбивает окружность.

Thus a chord divides a circle into two segments.

Таким образом, хорда разбивает окружность на два сегмента.

If these segments are equal, each is called a semicircle.

Если эти сегменты равны, они называются полуокружностями.

The segments are equal when the chord passes through the centre of the circle, i.e. is a diameter.

Сегменты равны, если хорда проходит через центр окружности, т.е. является диаметром.

Hence a semicircle is a plane figure bounded by a diameter and one of the arcs into which the diameter divides the circumference.

Таким образом, полуокружность – это часть плоскости, ограниченная диаметром и одной из дуг, которые образуются при разбиении окружности диаметром.

If the segments are unequal, the greater is called the major segment, and the less, the minor segment.

Если сегменты не равны, больший сегмент называется большим сегментом, а меньший – малым сегментом.

The arc of a major segment is a major arc; the arc of a minor segment is a minor arc.

Дуга большого сегмента – большая дуга; дуга малого сегмента – малая дуга.

Упражнение 16. Заполните пропуски предлогами.

1. The interior … (of) a circle C is the set… (of) all the points … (of) the plane… (of) the circle… (in) the distance less than r… (of) C.
2. It follows immediately… (no preposition) the definition that each point … (of) the plane … (of) a circle is… (no preposition) exactly one…. (of) the following sets: the interior… (of) the circle, the circle itself, the exterior… (of) the circle.
3. … (with) the help… (of) three lemmas it is possible to prove the following useful facts… (about) circles.
4. The distance… (from) the centre… (of) a circle… (to) a chord is the distance … (between) the centre and the line containing the chord.
5. … (in) a circle or… (in) two circles… (of) equal radii, two chords are equidistant…(from) the centre if and only if the chords are congruent.
6. A semicircle is a plane figure bounded… (by) a diameter and one… (of) the arcs…(into) which the diameter divides the circumference.
7. Four or more points… (on) which a circle may be drawn are said to be concyclic (лежащий на одной окружности).
8. The straight line… (from) the centre… (of) a circle… (to) the midpoint… (of) a chord, which does not pass… (through) the centre, is perpendicular… (to) the chord.
9. A straight line cannot have more than two points… (no preposition) common….(with) the circumference… (of) a circle.
10. This figure shows us the relation… (between) the length … (of) a chord and its distance… (from) the centre… (of) a circle.

Упражнение 17. Замените прямые вопросы косвенными по образцу:

Do all radii of the same circle have the same measure? – I wonder (I’d like to know) if all radii have the same measure.

1. What is the unit measure used in measuring an angle?

I wonder what unit measure is used in measuring an angle.

2. What does each angle measure if a triangle is equilateral?

I wonder what each angle measures if a triangle is equilateral.

3. What is the difference between a diameter and a radius?

I wonder what the difference between a diameter and a radius is.

4. What is an arc?

I wonder what an arc is.

5. What is another name for a circumference?

I wonder what another name for a circumference is.

6. By what formula do you express the ratio of the circumference © to a diameter (d)?

I wonder by what formula you express the ratio of the circumference © to a diameter (d).

7. What is the length of a diameter equal to?

I wonder what the length of a diameter is equal to.

8. Is c/2r the same for all circles?

I wonder if c/2r is the same for all circles.

9. Does a circle contain any segment? Why not?

I wonder if a circle contains any segment and why the circle doesn’t contain any segment.

Упражнение 18. Сделайте предложения более короткими, используя причастия вместо придаточных предложений.

1. After he had drawn a figure which served him as a visual aid he solved the problem very quickly.

Having drawn a figure serving him as a visual aid he solved the problem very quickly.

2. Do you know the theorem which was proved by Nikitin in a very original way?

Do you know the theorem proved by Nikitin in a very original way?

3. A circle is a figure which is bounded by the curved line all points of which are the same distance away from the fixed point, which is called the centre.

A circle is a figure bounded by the curved line all points of which are the same distance away from the fixed point called the centre.

4. The straight line that passes through the centre and both of whose ends are on the circumference is called a diameter.

The straight line passing through the centre and with both ends on the circumference is called a diameter.

5. A line segment that connects any two points on the circumference of a circle is a chord.

A line segment connecting any two points on the circumference of a circle is a chord.

6. A straight line that is drawn from the centre to any point on the circumference is called a radius.

A straight line drawn from the centre to any point on the circumference is called a radius.

Упражнение 19. Закончите предложения, используя чертеж.

а)

1. MN is called a…. of a circle. (chord)

2. MN is called a ….(segment)

3. MRN is a …. angle. (central)

4. MRK is a… angle. (central)

5. RM is called a … of a circle. (radius)

6. KN is twice as long as… (KR)

7. M is called a …. of a circle. (point)

8. RL has the same length as…(RK)

9. MRN is an… triangle. (isosceles)

b)

1. No matter how short an arc is it is… at least slightly. (longer than a chord)

2. The term circumference means…(a set of points in a plane each of which is equidistant)

3. A diameter is a chord which…(passes through the centre of the circle)

4. A circle is a set of points in a plane each of which…(is equidistant)

5. We cannot find the circumference of a circle by adding…(the lengths of its arcs).

Упражнение 20. Переведите следующие предложения на английский язык.

1. Отрезок прямой, соединяющий две какие-либо точки окружности, называется хордой.

A line segment joining any two points of a circle is called a chord.

2. Хорда, проходящая через центр окружности, называется диаметром.

A chord which passes through the centre of the circle a called a diameter.

Диаметр равен сумме двух радиусов и потому все диаметры одной окружности равны между собой.

The diameter is a sum of two radii. Hence all the diameters of the same circle are equal to each other.

Любая часть окружности называется дугой.

Any part of a circle is called an arc.

Упражнение 21. Задайте вопросы по содержанию текста.

Упражнение 22. Перескажите кратко текст на английском языке.