The length of the segment and its measurement. The length of a line segment and its measurement What is the length of a line segment

REPEATING THEORY

16. Fill in the blanks.

1) Point and line are examples of geometric shapes.
2) To measure a segment means to calculate how many unit segments it fits.
3) If point C is marked on the segment AB, then the length of the segment AB is equal to the sum of the lengths of the segments AC + CB
4) Two segments are called equal if they match when overlaid.
5) Equal segments have equal lengths.
6) The distance between points A and B is the length of the segment AB.

SOLVING PROBLEMS

17. Mark the segments shown in the figure and measure their lengths.

18. Draw all possible line segments with ends at points A, B, C and D. Write down the designations of all drawn line segments.

AB, BC, CD, AD, AC, BD

19. Write down all the segments shown in the picture.

20. Draw segments CK and AD so that CK \u003d 4 cm 6 mm, AD \u003d 2 cm 5 mm.

21. Draw a segment BE, the length of which is 5 cm 3 mm. Mark point A on it so that BA \u003d 3 cm 8 mm. How long is AE?

AE \u003d BE-VA \u003d 5cm 3mm - 3cm 8mm \u003d 1cm 5mm

22. Express the given quantity in the specified units.

23. Write down the links of the broken line and measure their length (in millimeters). Calculate the length of the polyline.

24. Mark point B, located 6 cells to the left and 1 cell below point A; point C, located 3 cells to the right and 3 cells below point B; point D, located 7 cells to the right and 2 cells above point C. Connect successively points A, B, C and D.

A broken line ABCD, consisting of 3 links, was formed.

25. Calculate the length of the polyline shown in the figure.

a) 5 * 36 \u003d 180 mm
b) 3 * 28 \u003d 84 mm
c) 10 * 10 + 15 * 4 \u003d 160 mm

26. Construct a polyline DSEK so that DС \u003d 18 mm, CE \u003d 37 mm, EK \u003d 26 mm. Calculate the length of the polyline.

27. It is known that AC \u003d 17 cm, BD \u003d 9cm, BC \u003d 3 cm. Calculate the length of the segment AD.

28. It is known that MK \u003d KN \u003d NP \u003d PR \u003d RT \u003d 3 cm. What other equal segments are there in this figure? Find their lengths.

29. Points were marked on the straight line so that the distance between any two adjacent points is 4 cm, and between the extreme points - 36 cm. How many points are marked?

30. Draw, without lifting the pencil from the paper, the figures shown in the figure. Each line can only be drawn with a pencil once.

If you touch a notebook sheet with a well-sharpened pencil, a trace will remain, which gives an idea of \u200b\u200bthe point. (fig. 3).

Let's mark two points A and B on a piece of paper. These points can be connected with different lines (fig. 4). How do you connect points A and B with the shortest line? This can be done with a ruler (Fig. 5). The resulting line is called segment.

Point and line - examples geometric shapes.

Points A and B are called the ends of the segment.

There is only one segment, the ends of which are points A and B. Therefore, the segment is denoted by writing down the points that are its ends. For example, the segment in Figure 5 is denoted in one of two ways: AB or BA. Read: "segment AB" or "segment BA".

Figure 6 shows three lines. The length of the segment AB is 1 cm. It is placed in the segment MN exactly three times, and in the segment EF exactly 4 times. We will say that segment length MN is 3 cm and EF is 4 cm.

It is also customary to say: "the segment MN is equal to 3 cm", "the segment EF is equal to 4 cm". They write: MN \u003d 3 cm, EF \u003d 4 cm.

We measured the lengths of the segments MN and EF single segment, the length of which is 1 cm.To measure the segments, you can choose other units of length, for example: 1 mm, 1 dm, 1 km. In Figure 7, the segment length is 17 mm. It is measured by a single line segment, the length of which is 1 mm, using a ruler with divisions. Also, using a ruler, you can build (draw) a segment of a given length (see Fig. 7).

At all, to measure a segment means to count how many unit segments it fits.

The length of a segment has the following property.

If point C is marked on segment AB, then the length of segment AB is equal to the sum of the lengths of segments AC and CB(fig. 8).

They write: AB \u003d AC + CB.

Figure 9 shows two segments AB and CD. These segments will overlap.

Two line segments are said to be equal if they coincide when superimposed.

Hence the segments AB and CD are equal. They write: AB \u003d CD.

Equal segments have equal lengths.

Of the two unequal segments, we will consider the largest one with the longer length. For example, in Figure 6, the EF segment is larger than the MN segment.

The length of the segment AB is called distance between points A and B.

If you arrange several segments as shown in Figure 10, you get a geometric figure, which is called broken line... Note that all the segments in Figure 11 do not form a polyline. The segments are considered to form a broken line if the end of the first segment coincides with the end of the second, and the other end of the second segment coincides with the end of the third, etc.

Points A, B, C, D, E - polyline vertices ABCDE, points A and E - polyline ends, and the segments AB, BC, CD, DE are its links (see fig. 10).

Polyline length call the sum of the lengths of all its links.

Figure 12 shows two broken lines, the ends of which coincide. Such broken lines are called closed.

Example 1 ... Segment BC is 3 cm less than segment AB, which is 8 cm long (Fig. 13). Find the length of the line segment AC.

Decision. We have: BC \u003d 8 - 3 \u003d 5 (cm).

Using the segment length property, we can write AC \u003d AB + BC. Hence AC \u003d 8 + 5 \u003d 13 (cm).

Answer: 13 cm.

Example 2 ... It is known that MK \u003d 24 cm, NP \u003d 32 cm, MP \u003d 50 cm (Fig. 14). Find the length of the line segment NK.

Decision. We have: MN \u003d MP - NP.

Hence MN \u003d 50 - 32 \u003d 18 (cm).

We have: NK \u003d MK - MN.

Hence NK \u003d 24 - 18 \u003d 6 (cm).

Answer: 6 cm.

The concept of the length of a segment and its measurements have already been used repeatedly, in particular, when a natural number was considered as a measure of magnitude. In this section, we will only generalize the concept of the length of a segment as a geometric quantity.

In geometry, length is a value that characterizes the length of a segment, as well as other lines (broken line, curve). In our course, only the concept of segment length will be considered. In defining it, we will use the notion “a segment consists of segments” introduced in Topic 18.

Definition.The length of a segment is a positive quantity that has the following properties: 1) equal segments have equal lengths; 2) if a segment consists of two segments, then its length is equal to the sum of the lengths of its parts.

These line length properties are used when measuring it. To measure the length of a segment, you need to have a unit of length. In geometry, such a unit is the length of an arbitrary segment.

As shown in topic 18, the result of measuring the length of a segment is a positive real number - it is called numerical value segment length with the selected unit of length or measure of length of this segment. If we denote the length of the segment with the letter X, the unit of length - E, and the real number obtained during the measurement with the letter a, then we can write: a \u003d m E (X) or X \u003d a ∙ E.

The positive real number obtained when measuring the length of a segment must satisfy a number of requirements:

1. If two segments are equal, then the numerical values \u200b\u200bof their lengths are also equal.

2. If the segment x consists of segments x 1 and x 2, then the numerical value of its length is equal to the sum of the numerical values \u200b\u200bof the lengths of the segments x 1 and x 2.

3. When changing a unit of length, the numerical value of the length of a given segment increases (decreases) as many times as the new unit is less (more) than the old one.

4. The numerical value of the length of a unit segment is equal to one.

It is proved that a positive real number, which is a measure of the length of a given segment, always exists and is unique. It is also proved that for every positive real number there is a segment whose length is expressed by this number.

Note that often, for the sake of brevity, the numerical value of the length of a segment is simply called length. For example, in the task “Find the length of a given segment” the word “length” means the numerical value of the length of the segment. Another liberty is no less often allowed - they say: “Measure the segment” instead of “Measure the length of the segment”.

A task. Construct a line segment with a length of 3.2E. What will be the numerical value of the length of this segment if the unit of length E is increased by 3 times?

Decision. Let's construct an arbitrary segment and consider it to be a unit. Then we construct a straight line, mark point A on it and set aside 3 segments from it, the lengths of which are equal to E. We obtain a segment AB, the length of which is 3E (Fig. 1).

To get a segment with a length of 3.2E, you must enter a new unit of length. To do this, the unit segment must be divided either into 10 equal parts, or into 5, since 0.2 \u003d. If from point B to postpone a segment equal to one, then the length of the segment AC will be equal to 3.2E.

To fulfill the second requirement of the problem, we will use property 3, according to which when the unit of length increases by 3 times, the numerical value of the length of a given segment decreases by 3 times. Divide 3.2 by 3, we get:

3.2: 3 \u003d\u003d 3: 3 \u003d \u003d 1. Thus, with a unit of length 3E, the numerical value of the length of the constructed segment AC will be equal to 1.

By segment called the part of a straight line consisting of all points of this line that are located between these two points - they are called the ends of the segment.

Let's look at the first example. Let a segment be given by two points in the coordinate plane. In this case, we can find its length using the Pythagorean theorem.

So, in the coordinate system, draw a segment with the given coordinates of its ends (x1; y1) and (x2; y2) ... On axis X and Y omit the perpendiculars from the ends of the segment. Mark in red the segments that are projections from the original segment on the coordinate axis. After that, we transfer the projection segments parallel to the ends of the segments. We get a triangle (rectangular). The segment AB itself will become the hypotenuse of this triangle, and the transferred projections are its legs.

Let's calculate the length of these projections. So on the axis Y the projection length is y2-y1 , and on the axis X the projection length is x2-x1 ... Let's apply the Pythagorean theorem: | AB | ² \u003d (y2 - y1) ² + (x2 - x1) ² ... In this case | AB | is the length of the line segment.

If you use this scheme to calculate the length of a segment, then you can even not build a segment. Now let's calculate what is the length of the segment with coordinates (1;3) and (2;5) ... Applying the Pythagorean theorem, we get: | AB | ² \u003d (2 - 1) ² + (5 - 3) ² \u003d 1 + 4 \u003d 5 ... This means that the length of our segment is 5:1/2 .

Consider the following way to find the length of a line segment. To do this, we need to know the coordinates of two points in some system. Consider this option using a two-dimensional Cartesian coordinate system.

So, in a two-dimensional coordinate system, the coordinates of the extreme points of the segment are given. If we draw straight lines through these points, they must be perpendicular to the coordinate axis, then we get a right-angled triangle. The original segment will be the hypotenuse of the resulting triangle. The legs of the triangle form segments, their length is equal to the projection of the hypotenuse on the coordinate axis. Based on the Pythagorean theorem, we conclude: in order to find the length of a given segment, you need to find the lengths of projections on two coordinate axes.

Find the lengths of the projections (X and Y) the original segment to the coordinate axes. We will calculate them by finding the difference in the coordinates of points along a separate axis: X \u003d X2-X1, Y \u003d Y2-Y1 .

Calculate the length of the segment A , for this we find the square root:

A \u003d √ (X² + Y²) \u003d √ ((X2-X1) ² + (Y2-Y1) ²) .

If our segment is located between points whose coordinates are 2;4 and 4;1 , then its length, respectively, is √ ((4-2) ² + (1-4) ²) \u003d √13 ≈ 3.61 .