Cut length and measurement. Length of segment and its measurement what is equal to the length of the length of the straight

We repeat the theory

16. Fill in skipping.

1) The point and segments are examples of geometric shapes.
2) Measure the segment means to calculate how many single segments are placed in it.
3) If on the segment AV overtake the point C, then the length of the segment Av is equal to the sum of the lengths of the segments of the AC +
4) two segments are called equal if they coincide when applied.
5) Equal segments have equal lengths.
6) the distance between the points A and B is called the length of the segment AB.

We solve the tasks

17. Recover the segments shown in the figure, and measure their lengths.

18. Perform all possible segments at the ends at the points A, B, C and D. Record the designations of all segments spent.

AB, B, CD, AD, AU, VD

19. Record all the segments depicted in the picture.

20. Instruct the segments of the SC and AD so that the SC \u003d 4 cm is 6 mm, AD \u003d 2 cm 5 mm.

21. Instruct the segment of ve, the length of which is 5 cm 3 mm. Mark on it a point and so that be \u003d 3 cm 8 mm. What is the length of the segment ae?

Ae \u003d ve-va \u003d 5 cm 3 mm - 3 cm 8mm \u003d 1 cm 5mm

22. Express this amount in the specified units of measurement.

23. Record the loloral links and measure their lengths (in millimeters). Calculate the length of the broken.

24. Mark the point B located on 6 cells to the left and 1 cell below the point A; The point C, located on 3 cells to the right and 3 cells below the point in; Point D, located on 7 cells to the right and 2 cells above the C point. Connect consistently by segments of points A, B, C and D.

Formed a broken AVD, consisting of 3 units.

25. Calculate the length of the broken depicted in the figure.

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

26. Build a broken DSE that DC \u003d 18 mm, CE \u003d 37 mm, EK \u003d 26 mm. Calculate the length of the broken.

27. It is known that ac \u003d 17 cm, cd \u003d 9cm, Sun \u003d 3 cm. Calculate the length of the AD segment.

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. On line noted the points so that the distance between two any adjacent points is 4 cm, and between the extreme dots - 36 cm. How many points is noted?

30. Inscribe, without opening a pencil from paper, figures depicted in the picture. Each line can be carried out with a pencil only once.

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

Note on a sheet of paper two points A and B. These points can be connected by various lines (Fig. 4). And how to connect points a and b the shortest line? This can be done with the help of the ruler (Fig. 5). The resulting line is called Cut.

Point and Cut - Examples geometric figures.

Points a and b call segments of cut.

There is a single segment whose ends are points A and B. Therefore, the segment is indicated by writing the points that are its ends. For example, a segment in Figure 5 is denoted by one of two ways: AB or BA. They read: "Cut AB" or "Cut BA".

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

It is also customary to say: "Mn segment is 3 cm", "EF segment is 4 cm." Writing: Mn \u003d 3 cm, EF \u003d 4 cm.

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

At all, measure the segment means to calculate how many single segments are placed in it.

The length of the segment has the following property.

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

Write: AB \u003d AC + CB.

Figure 9 shows two cuts of AB and CD. These segments are coincided with these segments.

Two segments are called equal, if they coincide when applied.

Therefore, the segments of AB and CD are equal. Write: ab \u003d CD.

Equal segments have equal lengths.

Of the two unequal segments, we will consider the one that the engine is more. For example, in Figure 6, the EF segment is larger than the Mn segment.

AB cut length is called distance between points A and B.

If several segments are positioned as shown in Figure 10, then the geometric figure is obtained, which is called loan. Note that all the segments in Figure 11 are not formed. It is believed that segments form a broken, if the end of the first segment coincides with the end of the second, and the other end of the second segment - with the end of the third, etc.

Points A, B, C, D, E - the vertices of the broken ABCDE, points a and e - ends of scratch, and segments AB, BC, CD, DE - it links (See Fig. 10).

Long broken Call the sum of the lengths of all its links.

Figure 12 shows two broken, the ends of which coincide. Such broken call closed.

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

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

Using the property of the length of the segment, you can write AC \u003d AB + BC. Hence the 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 NK segment.

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

Hence the Mn \u003d 50 - 32 \u003d 18 (cm).

We have: nk \u003d mk - Mn.

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

Answer: 6 cm.

The concept of the length of the segment and its measurements was already used repeatedly, in particular, when the natural number was considered as a measure of magnitude. In this paragraph, we only summarize the presentation of the length of the segment as a geometrical value.

In geometry length is a value that characterizes the length of the segment, as well as other lines (broken, curve). In our course only the concept of the length of the length of the segment will be considered. With its definition, we will use the concept of "segments of segments" introduced in the topic 18.

Definition.The length of the segment is called a positive value, which has the following properties: 1) Equal segments have equal lengths; 2) If the segment consists of two segments, then its length is equal to the sum of the lengths of its parts.

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

As shown in the topic 18, the result of measuring the length of the segment is a positive valid number - it is called numerical meaning cut lengths with a selected unit length or measure length This segment. If you designate the length of the segment of the letter x, the unit length is e, and the actual number obtained by measurement is the letter A, then it can be written: a \u003d m e (x) or x \u003d a ∙ e.

Received when measuring the length of the segment, a positive valid number should 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 segments x 1 and x 2.

3. When replacing the length of the length, the numerical value of the length of this segment increases (decreases) to as many times as the new one (more) old.

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

It has been proven that a positive valid number, which is a measure of the length of a given segment, always exists and is unique. It is also proved that for each positive actual number there is a segment whose length is expressed by this number.

Note that often, for the sake of brevity speech, the numerical value of the length of the segment is called simply length. For example, in the task "Find the length of this segment" under the word "length" means the numerical value of the length of the segment. No less frequently admit another library - they say: "Measure the segment" instead of "measuring the length of the segment".

A task. Build a segment whose length is 3.2. What will be the numerical value of the length of this segment, if the unit of length E is 3 times?

Decision. We construct an arbitrary segment and we will consider it a single one. Then we will construct a straight line, we note on it a point A and postpone from it 3 segments whose lengths are equal to E. We obtain a segment of the AB, the length of which is 3e (Fig. 1).

To get a length of 3,2, it is necessary to introduce a new unit of length. For this, a single segment must be broken either by 10 equal parts or 5, since 0.2 \u003d. If from the point to postpone the segment equal to a single, then the length of the speaker is equal to 3,2.

To fulfill the second requirement of the task, we use the property 3, according to which, with an increase in the length of the length, 3 times the numerical value of the length of this segment is reduced by 3 times. We divide 3.2 to 3, we get:

3.2: 3 \u003d\u003d 3: 3 \u003d \u003d 1. Thus, when the length of the length is 3E, the numerical value of the length of the constructed spell will be equal to 1.

Cut They call a part of a straight line consisting of all points of this line, which are located between the data with two points - they are called the sections of the segment.

Consider the first example. Suppose in the plane of coordinates set by two points a certain segment. In this case, we can find it, using Pythagore's theorem.

So, in the coordinate system, draw a segment with the specified coordinates of its ends (x1; y1) and (x2; y2) . On axis X. and Y. From the end of the segment to omit the perpendicular. We note the red segments that are on the axis of coordinate projections from the initial segment. After that, we move in parallel to the ends of the segments of the projection segment. We get a triangle (rectangular). The hypotenurus of this triangle will be the segment of AB, and its categories are transferred projections.

Calculate the length of these projections. So, on the axis Y. The length of the projection is equal y2-Y1. , and on the axis H. The length of the projection is equal x2-x1. . Apply Pythagore's theorem: | AB | ² \u003d (y2 - y1) ² + (x2 - x1) ² . In this case | AB | is a length of segment.

If you use this scheme to calculate the length of the segment, you can even cut and not build. Now calculate what the length of the segment with coordinates (1;3) and (2;5) . Using the Pythagora 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 equal 5:1/2 .

Consider the following method of finding the length of the segment. To do this, we need to know the coordinates of two points in any system. Consider this option by applying 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 you spend direct lines through these points, they must be perpendicular to the axis of the coordinates, then we get a rectangular triangle. The initial segment will be a hypothenose of the resulting triangle. Triangle kartets form segments, their length is equal to the projection of hypotenuses on the axis of coordinates. Based on the Pythagoreo Theorem, we conclude: in order to find the length of this segment, you need to find the lengths of projections into two axes of coordinates.

Find lengths of projections (X and y) The initial segment on the coordinate axes. They will calculate them by finding the difference in the coordinates of points on a separate axis: X \u003d x2-x1, y \u003d y2-y1 .

Calculate the length of the cut BUT For this we will find a square root:

A \u003d √ (x² + y²) \u003d √ ((x2-x1) ² + (y2-y1) ²) .

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