Points and Lines

The world around us contains many physical objects from which mathematicians have developed geometric ideas and 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 mine straight line. A geometric line is a 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, 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 say of the models of points as point A, point B, and point C. We say of the lines as a line AB or line BA for example.

As we know, between any two points on a line there is another point. So, a line consists of a set of points. Therefore, a piece of the line is a subset of the line or a line segment in other words. A line segment is a set of points consisting of the two endpoints and all of the points on the line between them. How does a line segment differ from a line? A line segment has definite length but a line extends indefinitely in each of its two direction.

Another important subset of a line is called a ray. A ray has infinite length and only one endpoint which is called a vertex. How to mark a line, a line segment, or a ray in writing?

Task 17

Compare the ing-forms in these sentences and translate the sentences.

1. Measuring land is impossible without special instruments. 2. Measuring length of a segment one must use a ruler. 3. He is defining the volume of a geometric object. 4. The teacher spoke of defining volumes. 5. Geometry presented practical ways for obtaining information about the size and shape of various objects. 6. Obtaining that information we shall be able to extend our knowledge of space. 7. Extending a line indefinitely can be represented symbolically with the help of arrows indicating opposite direction. 8. Now I am extending the given line both to the left and to the right. 9. We use these figures for eliminating ambiguities. 10. Knowing the length and the width of an object is necessary for finding its volume. 11. Physical objects consist of particles moving about. 12. Emphasizing this fact is very important.

Task 18

Study the vocabulary:

To investigate – досліджувати;

property – властивість;

earth – земля;

measure – міра, вимірювати;

practical application – практичне застосування;

dimension – вимір;

assumption – припущення.

Task 19

Read the text "The history of geometry" and do the true-false test:

1. Only engineers and architects use lines and figures in their daily work.

2. Geometry is the branch of mathematics which investigates the relations, properties, and measurement of solids, surfaces, lines and angles.

3. Geometry is the Greek name for the science which the early Egyptians began and developed about 100 years ago.

4. For building pyramids the early Greeks needed professional geometers who were able to locate a line running north and south.

5. The Egyptians were principally interested in the practical application of their rules.

6. After a time Greek philosophers and teachers developed and perfected the proofs of the Egyptians.

7. Euclid was a teacher of geometry in Italy

8. Pythagoras and Plato were good friends and partners.

9. In the 19^th century the Russian mathematician Lobachevsky founded non-Euclidean geometry of two dimensions.

10. Riemann assumes that no straight line can be drawn which will not meet any other straight line.