With such a form, like a trapezium, we often do it in life. For example, be it a place, a kind of viking from concrete blocks, we’ll cut it with a butt. The best option is to use kermo-skin transport protection and more. About the power of the figure was known in Ancient Greece, describing in more detail Aristotle in his scientific practice “The Cob”. I knowledge, learned thousands of years ago, is relevant to this day. So let's get to know them in more detail.

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Basic understanding

1. The classic shape of the trapezium.

A trapezium for its daylight, which is made up of two winds, which are parallel, and two others, which are not parallel. When speaking about this figure, it is necessary to keep in mind the same understanding: the base, the height and the middle line. Two arms of a chotirikutnik, like one to one, are called bases (arms of AD and BC). The height is the name given to the height perpendicular to the skin base (EH), tobto. folded under a 90 ° cut (as shown in Fig. 1).

If you add up all the degrees of the inner ones, then the sum of the trapezes is more expensive 2π (360 °), like a chotirikutnik. Vіdrіzok, kіnci є є middle sidewalls (IF) call the middle line. Dovzhina tsgogo vіdrіzka to become the sum of substav BC and AD delenu by 2.

There are three kinds of geometric figures: straight, sizable and evenly flanked. If you want one kut at the tops of the base, it will be straight (for example, if ABD \u003d 90 °), such a cut is called a straight trapezium. Yakshcho bіchnі vіdіzki rіvnі (AB і CD), won is called isosceles (vіdpovіdno kuti with podstava іvnі).

How to know the area

For that, to know the square of the Chotirikutnik ABCD is corroded by the following formula:

Figure 2. Distribution of tasks on the search area

For a scientific butt, it’s easier to get started. For example, let the upper and lower bases be equal 16 and 44 cm long, and the sides - 17 and 25 cm. We must admit that

Come on DF - be. Z ΔADE (which will be even-sided), take it step by step:

Tobto, hanging by simple mine, on the back knew the height of ΔADE, like behind the madness є and the height of the trapezium. Zv_dsi are calculated according to the given formula for the area of ​​​​the chotirikutnik ABCD, from the given values ​​​​of the height DF.

Zvіdsi shukana area ABCD dovnyuє 450 cm³. Tobto it is possible to say with confidence that in order in order to calculate the area of ​​the trapezium, it is only necessary to add the sum of money to the height of the dove.

Important! If the order is not obov'yazkovo to know the value of dozhin okremo, it is generally allowed, as there will be zastosovannye іnshі parameters of the figure, yakі for vydpovidnogo proof dоrіvnyuvatimut sum_ pіdstav.

See the trapeze

Fallow in addition, as the sides of the maє figure, as the cuts are made on the bases, three types of chotirikutnik are seen: straight, raznoboka and ravnoboka.

Riznoboka

Use two forms: gostrokutna and stupid. ABCD is a state-of-the-art only in that case, if it is at the base (AD) of hospitality, and on the other hand, it is different. If the value of one kuta number Pi / 2 is greater (the degree of the world is greater for 90 °), then we take it stupidly.

Yakshcho sidewall on dozhina rіvnі

Figure 3. View of a trapezoid

Even if the non-parallel sides are equal in length, then ABCD is called equal-sided (correct). At the same time, in such a chotirikutnik, the degree of the world is the same when presented, their kut is less than the direct one. The very same causes of rіvnofemoral in no way can be divided into gostrokutnі and obtuse. Chotirokhkutnik of such a form may have its own specific powers, to which it can be carried:

1. Vіdrіzki, scho zadnuyut protilezhnі peaks, equal.
2. Gostrі kuti with a larger prodstavі become 45 ° (the front butt of the little 3).
3. If you fold the degrees of the protilazhny kutіv, then the sum of the stench will give 180 °.
4. How much you can induce to be the right trapeze.
5. How to fold the degree world of the adjacent kutіv, won't do it π.

Moreover, through its geometrical expansion, the point is found main power of the equal-femoral trapezium:

The value of the cut on the base is 90°

Perpendicularity of the side of the base - this is the characteristic of the concept of "rectangular trapezium". Two lateral sides with kutas on the stand can not be, to that, otherwise it will be a straight cut. Chotirikutniks of this type of friend have a wicked side, which always establishes a gostry kut with a great foundation, and with a smaller one, a dull one. With which, the perpendicular side will also be tall.

Vіdrіzok between the middles of the sidewalls

If the backs of the middle of the side walls, and the omissions of the arms will be parallel to the bases, and dove by the same half of their sumi, then a straight line is established be the middle line. The value of the price is calculated according to the following formula:

For the night butt, we can look at the task from the stasis of the middle line.

Manager. The middle line of the trapezium is 7 cm long, it seems that one of the sides is 4 cm longer than the other (Fig. 4). Know the basics.

Figure 4

Solution. Let the smaller base of DC be stronger x cm, then the larger base is more dense (x + 4) see.

Come out, the smaller base of the DC is 5 cm, and the larger is 9 cm.

Important! The concept of the middle line is the key to the hour of completion of the bagatioh problem of geometry. On the basis of this appointment, there will be a lot of evidence of other figures. Vikoristovuyuchi understanding in practice, it is possible to have a greater rational solution to that need of a necessary magnitude.

Appointed heights and ways to know how її know

As it was conceived earlier, the height is є vіdrіzok, which peretinaє pіdstavi pіd kutom 2Pi / 4 and є the shortest vіdstannyu z-pomіzh them. Before it, how to know the height of the trapezium, next to indicate the given input values. For the best understanding, let's look at the task. Know the height of the trapezium for the mind, which bases are 8 and 28 cm, the sides are 12 and 16 cm, obviously.

Figure 5

Let's make cuts DF and CH with straight cuts to the base of AD. Vіdpovіdno up to the appointment, the skin from them will be the height of the given trapezії (Fig. 5). In this case, knowing the length of the skin sidewall, with the help of the Pythagorean theorem, we know why the height in AFD and BHC knitwear is good.

The amount of money in AF and HB is more expensive than the basics, tobto:

Let AF dovzhina more x cm, then do dozhina vіdrіzka HB=(20 – x) div. How it was installed, DF = CH, stars.

Then we take away the next equal:

It turns out that the AF tresses in the AFD tricoutnik are more than 7.2 cm, we can calculate the height of the trapezium DF according to the same Pthagorean theorem:

Tobto. the height of the ADCB trapezium is 9.6 cm. Ale, for a number of tasks with geometry, they can only have more degrees of cut, in such a case, the calculations will be carried out through the spivvіdnoshennia of the sides of the internal trikutniks.

Important! In essence, a trapezoid is often seen as two tricouts, or as a combination of a rectangle and a tricout. For the sake of 90% of all tasks, which are heard by school assistants, power and signs of these figures. Most of the formulas, in case of GMT, rely on "mechanisms" for assigning two types of figures.

Yak shvidko calculate the dozhina of the foundation

Before that, in order to know the basis of the trapeze, it is necessary to determine, as the parameters are already given, and as a rational choice. A practical approach is to improve the old unknown foundation of the formula of the middle line. For a clearer picture of the picture, it is shown from the butt of the task, how and to work. Let's see that the middle line of the trapezium is 7 cm, and one of the bases is 10 cm. Know the length of the other base.

Solution: Knowing that the middle line is more than half of the sum of the basics, it is possible to confirm that the sum of more is 14 cm.

(14cm=7cm×2). Think about it, we know that one is 10 cm, the smaller side of the trapezium is more than 4 cm (4 cm \u003d 14 - 10).

More than that, for a more comfortable execution of such a plan, it is recommended to kindly use such formulas from the trapezium area as:

• middle line;
• area;
• height;
• diagonals.

Knowing the essence (the very essence) of these can be calculated without special zusil to recognize the meaning of shukane.

Video: trapeze and power

Video: features of the trapeze

Visnovok

From looking at the butts of the task, you can make simple visnovok, like a trapezium, like a calculation of the task, is one of the simplest figures of geometry. For a successful outcome, the task for everything is not assigned to it, as information about the described object, in some formulas it can be stuck, and assigned to it, it is necessary to know. Using this simple algorithm, there is a similar problem of zastosuvannya tsієї geometrical figure not in stock zusil.

In the materials of various control studies and experiments, they often appear task for the trapeze, Virishennya yakikh vmagaє knowledge of її authorities.

It is clear that the trapezium is a trapezoid for the accomplishment of the tasks of authority.

After the increase in the power of the middle line of the trapezium, it is possible to formulate that the power of the cross, what happens to the middle of the diagonals of the trapezium. Vіdrіzok, scho z'єdnuє middle of the diagonals of the trapezium, dorіvnyuє vіvіrіznosti basics.

MO - middle line of tricot ABC and 1/2BC (Fig. 1).

MQ - the middle line of the tricot ABD is 1/2AD.

Also OQ = MQ - MO, also, OQ = 1/2AD - 1/2BC = 1/2(AD - BC).

When executing a bagatioh, a task for a trapeze with one of the main methods is carried out at two heights.

Let's take a look manager.

Nehai BT - the height of the equal-femoral trapezium ABCD with bases BC and AD, moreover, BC = a, AD = b. Know the dozhini vіdrіzkіv AT and TD.

Solution.

The accomplishment of the task does not call out the difficulties (Fig. 2) ale vono allow otrimati the power of the height of the equal-femoral trapezium, drawn from the top of the blunt kut: the height of the equal-femoral trapezium, drawn from the top of a blunt kut, to divide the larger base into two ribs, the smaller one from some more beautifully folded bases, and the larger one - to sum up the bases.

When the power of the trapeze has been vindicated, there has been an increase in respect for such power, like a similarity. So, for example, the diagonals of the trapezium are divided into chotiri tricouts, moreover, the tricouts, which lie to the bases, are similar, and the tricouts, which lie to the sides, are equal in size. Tse firmness can be called the power of trikutniks, on which the trapezium is broken with diagonals. Moreover, the first part of hardening can be easily brought through the sign of the similarity of trikutniks from two cuts. Brought to you another part of the hardened.

Tricks BOC and COD (Fig. 3) yakscho accept for їх submissions BO and OD. Then S BOC / S COD = BO / OD = k. Also, S COD = 1/k S BOC.

Similarly, the tricots BOC and AOB can be cast high, so they can be taken for their representations of the CO and OA. Then S BOC / S AOB \u003d CO / OA \u003d k and S A O B \u003d 1 / k S BOC.

Three of these two propositions are clear that S COD = S A O B.

Let's not dwell on the formulated firmness, but let's know a link between the squares of trikutniks, on which a trapezoid is broken with diagonals. For whom we see such a task.

Let the point O be the point of crossing the diagonals of the trapezium ABCD from the bases BC and AD. Apparently, the area of ​​tricots BOC and AOD is equal to S1 and S2. Know the area of ​​the trapezium.

Scod S COD \u003d S A O B, then S ABC D \u003d S 1 + S 2 + 2S COD.

Z similar trikutnikiv BOC and AOD vyplivaє that VO / OD \u003d √ (S₁ / S 2).

Also, S₁/S COD = BO/OD = √(S₁/S 2), also S COD = √(S 1 S 2).

Then S ABC D \u003d S 1 + S 2 + 2 √ (S 1 S 2) \u003d (√ S 1 + √ S 2) 2.

To the victories of similarity to be brought and the power of the cross, to pass through the cross point of the diagonals of the trapezium parallel to the bases.

Look at manager:

Let the point O - the point of the crossbar of the diagonals of the trapezium ABCD іz with the bases BC and AD. BC=a, AD=b. Find the length of the crossbar PK, which is to pass through the cross point of the diagonals of the trapezoid parallel to the bases. How are the windings divided by the PK point O (Fig. 4)?

Z similar trikutnikov AOD and BOC vyplivaє, shcho AO / OS = AD / BC = b / a.

Z similar to tricots AOR and ACB is obvious, so AO / AC \u003d PO / BC \u003d b / (a ​​+ b).

Stars PO = BC b / (a ​​+ b) = ab / (a ​​+ b).

Similarly, similar to DOK and DBC, we see that OK = ab/(a + b).

Stars PO = OK and PK = 2ab/(a + b).

Later, the authority can be formulated as follows: a cross, parallel to the bases of the trapezium, to pass through the point of the crossbar of the diagonals and the rear two points on the sides, to share the point of the crossbar of the diagonals navpil. Yogo dozhina is the middle harmonic basis of the trapezium.

Step on the power of some point: at the trapezium, the point of the crossbar of the diagonals, the point of the crossbar of the continuation of the side sides, the middle of the bases of the trapezium lie on the same line.

Tricots BSC and ASD similar (Fig. 5) and in the skin of them, median ST and SG divide at the top of S on the same part. Also, points S, T and G lie on the same straight line.

So it is on one straight line of the points T, O and G. This is similar to tricots BOC and AOD.

Also, all points S, T, O and G lie on the same straight line.

So you yourself can know the length of the trapezium, which breaks, into two similar ones.

Like trapezium ALFD and LBCF similar (Fig. 6), then a/LF = LF/b.

Stars LF = √(ab).

In such a rank, vіdrіzok, scho breaking the trapezium into two similar trapezії, maє dozhina equal to the average geometric dozhin of the foundations.

Brought to you power in the air, what to divide the trapezium on two equally large.

Let the area of ​​the trapezium be more beautiful S (Fig. 7). h 1 і h 2 - Parts of the height, and x - Dovzhina shukany v_drіzka.

Then S / 2 \u003d h 1 (a + x) / 2 \u003d h 2 (b + x) / 2 that

S \u003d (h 1 + h 2) (a + b) / 2.

Warehouse system

(h 1 (a + x) = h 2 (b + x)
(h 1 (a + x) = (h 1 + h 2) (a + b)/2.

Considering the system, we take x \u003d √ (1/2 (a 2 + b 2)).

in such a manner, dovzhina v_drіzka, scho to divide the trapezium into two equally large, dorіvnyuє √ ((a 2 + b 2) / 2)(Mean quadratic dozhin fundamentals).

Also, for the trapezium ABCD with the bases AD and BC (BC = a, AD = b) they brought it to the top:

1) MN, which is behind the middle of the lateral sides of the trapezium, parallel to the bases and dovnyu їkh napіvsumі (arithmetic mean of numbers a and b);

2) PK, to pass through the cross point of the diagonals of the trapezium parallel to the bases, more
2ab/(a + b) (to the harmonic mean of numbers a and b);

3) LF, which divides the trapezoid into two similar trapeziums, making the difference between the geometric mean numbers a and b, √(ab);

4) EH, how to divide the trapezoid into two equally large ones, let's do it √ ((а 2 + b 2)/2) (mean square of the numbers a and b).

The sign of that power is inscribed and described trapezium.

The power of the inscribed trapezium: the trapezium may be inscribed in the colo in the same way, if it is even-femured.

The dominance of the described trapezium. If you can describe the trapezium and then, if the sum of the dozhins of the foundations is equal to the sum of the dozhins of the side sides.

Corinthian traces of what is inscribed in the trapezium:

1. The height of the described trapezium is equal to two radii of the inscribed stake.

2. The lateral side of the described trapezium is visible from the center of the inscribed stake under a straight apex.

The first is obvious. To prove another consequence, it is necessary to establish that COD is direct, so that one does not become a great practice. Then the knowledge of this consequence allows you to complete the task of vicoristing a straight-cut tricutnik.

Concretized consequences for the equifemoral description of the trapezium:

The height of the equal-femoral described trapezium is the middle geometric of the trapezium bases.
h = 2r = √(ab).

Perceived by the powers that be allow to know the trapezium better and ensure the success of the higher order of the stagnation of the powers.

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In these statutes, it is possible to replicate the power of the trapezius in every way. Zocrema, let's talk about the signs of power and the power of the trapezium, as well as about the power of the inscribed trapezium and about the colo, inscribed in the trapezium. We cherish mi and dominance of a rіvnofemoral and rectangular trapezium.

The butt of solving tasks for the best of all the powers that be looked at will help you lay out the places at the head and better remember the material.

Trapeze and all-all-all

For the cob, we briefly guess what such a trapezium and how to understand is connected with it.

Also, the trapezium is a figurine-chotirohkutnik, two sides of which are parallel to one to one (substantiate). І dvі not parallel - ce bіchnі sides.

At the trapezium, the height may be lowered - the perpendicular to the bases. Drawn the middle line and diagonals. And also, from whatever cut of the trapezium, you can draw a bisectrix.

Let's talk about the differences in power, connected with us with these elements and combinations.

Dominance of the diagonals of the trapezium

In order to be wiser, while reading, throw a trapezium ACME on the sheet and draw it at the diagonal.

1. Yakshcho you will know the middle of the skin from the diagonals (significantly qi points X and T) and you will know them, you will see the vіdіzok. One of the dominance of the diagonals of the trapezium is for the one whose vertex of the CT lies on the middle line. And yoga dozhina can be otrimavshi, dividing the difference on the two: XT \u003d (a - b) / 2.
2. Before us is the very trapezium ACME. The diagonals are tinted at the point O. Let's take a look at the AOE and IOC tricouts, made with diagonals cut together from the bases of the trapezium. Tsі trikutniks - similar. The similarity coefficient of tricots is expressed through the extension of the basics of the trapezium: k = AE/KM.
   The area of ​​tricots AOE and IOC is described by the coefficient k 2 .
3. All the same trapezium, those diagonals, which intertwine in the points O. Only once can we see the tricots, like the triangles of the diagonals, they have merged together with the side sides of the trapezium. The areas of trikutniks AKO and EMO are equally large - their areas are the same.
4. Another power of the trapezium includes the presence of diagonals. So, if you continue the lateral sides of AK and MO on a straight line with a smaller base, then it’s too early for the stench to flicker to the singing point. Far, through the middle of the bases of the trapezium, we draw a straight line. Vaughn is changing the foundations at the points X and T.
   Now we can continue the straight XT, there at once the point of the line of the diagonals of the trapezium O, the point, at which the line of the lateral sides and the middle of the lines X and T are intertwined.
5. Through the cross point of the diagonals we draw a cross, which is the main base of the trapezium (T lies on the smaller base KM, X - on the larger AE). The point of the crossbar of the diagonals is to divide the streak at the offensive sp_v_dnoshnі: TO/OH = KM/AE.
6. And now, through the cross point of the diagonals, we draw a parallel to the supports of the trapezium (a and b) of the vіdrіzok. The point of the crossbar is divided into two equal parts. You can find out the value of the wind by using the formula 2ab/(a + b).

Dominance of the middle line of the trapezium

Draw the middle line at the trapezium in parallel to the її pedestal.

1. The length of the middle line of the trapezium can be calculated, so as to lay down the length of the foundations and divide them: m = (a + b)/2.
2. How to draw a trapezium through the obedience, whether it be a windrow (high, for example), divide the middle line into two equal parts.

The power of the bisector of the trapezium

Choose whether there is a cut of the trapeze and conduct a bisectrix. Let's take, for example, kut KAЄ our trapezії Akme. Vikonavshi pobudova independently, and easily perekonaetsya - bisektrisa vіdsіkaєtsya vіd basi (or yogo prodovzhennia on a straight line beyond the borders of the figure itself) vіdrіzok ї dovzhini, scho and bіchna side.

The dominance of the kutіv trapezії

1. Yaku was not chosen from two pairs of kutivs lying to the side of the side, the sum of the kutivs of the pair would be 180 0: α + β = 180 0 and γ + δ = 180 0.
2. Z'єdnaєmo in the middle of the bases of the trapezium with a vіdrіzk TX. Now let's marvel at the kuti at the bases of the trapezium. How much sum of kutivs, if any of them become 90 0 dovzhin vіdrіzka TX, it is easy to calculate the outliers from the difference of dovzhin pіdstav, divided navpіl: TX \u003d (AE - KM) / 2.
3. If you draw parallel straight lines through the sides of the trapezium, then separate the sides of the trapezoid into proportional ribs.

Dominance of the rіvnofemoral (rіvnolateral) trapezії

1. At the equal-femoral trapeze, equal cuts for whatever foundations.
2. Now wake up the trapeze again, so that it would be easier to show what the language is. Look respectfully at the base AE - the top of the protile base M is projected as a point on the straight line, so as to avenge AE. Stand in the top A to the point of projection of the top M and the middle line of the equifemoral trapezium - equal.
3. A piece of words about the density of the diagonals of a rіvnofemoral trapezії - їх dozhina rіvnі. And also, however, cut these diagonals to the base of the trapezium.
4. Only a little close to the equal-femoral trapezium can describe the colo, the shards of the sum of the protilezhny kutivs of the chotirikutnik 1800 - the obov'yazkov of the mind for this one.
5. From the front point, the power of a rіvnofemoral trapezії is evident - as if a trapezії can be described as a colo, there is a rіvnofemoral trapezium.
6. Of the features of the equal-femoral trapezium, the power of the height of the trapezium is exuberant: if the diagonals are tucked under a straight kut, then the height of the height is more than half the sum of the basics: h = (a + b)/2.
7. I want to draw a TX cross through the middle of the bases of the trapezium - at the equal-femoral trapezium of the veins with a perpendicular to the bases. І one hour ТХ – all symmetry of the rіvnofemoral trapezії.
8. How many times to lower on a larger basis (significantly її a) the height from the opposite top of the trapezії. Weide two windbreakers. You can know the dozhina of one, as if to put it together and split it up: (a+b)/2. The other is taken away, if from a larger basis it is seen less and the difference is divided into two: (a – b)/2.

The power of the trapezium, inscribed in the column

Since the language has already gone about being inscribed in the trapezium, let's go to that nutritional report. The problem is that where the center of the stake is located according to the length to the trapezium. Here it is also recommended not to hesitate, take the olive by the hand and christen those who are to be lowered. So, understand better, and remember it better.

1. The roztashuvannya to the center of the stake is marked with a cut of the diagonal of the trapezium to the її flank side. For example, the diagonal can go from the top of the trapezium under a straight tail to the flank side. In such a case, the center of the described stake is changed exactly in the middle (R = ½AE).
2. The diagonal and the flank side can be serrated and under the hospitable kut - the same center of the stake appears in the middle of the trapezium.
3. The center of the described stake can appear behind the borders of the trapezium, behind the great її base, as between the diagonal of the trapezium and the side of the side - a blunt kut.
4. Kut, making the diagonal and the great basis of the trapezium AKME (inscriptions of the kut) fold half of the central kut, which yoma inspires: GRAVNE = ½MOЄ.
5. Briefly about two ways to calculate the radius of the described stake. The first way: respectfully marvel at your armchair - what do you want? You can easily remember that the diagonal splits the trapezium into two tricots. The radius can be known through the extension of the side of the tricot to the sine of the protractile kut, multiplied by two. For example, R \u003d AE / 2 * sinAME. Similarly, the formula can be written for whether it is on the other side of both tricots.
6. There is another way: we know the radius of the described stake through the square of the tricot, set diagonally, with the side side and the base of the trapezium: R \u003d AM * ME * AE / 4 * S AME.

The dominance of the trapezium described by the stake

It is possible to enter a stake into a trapezium, so that only one mind can be reached. More details about it below. And at the same time, this combination of figures has a low power base.

1. As a colo is inscribed in the trapezium, you can easily know the length of the її middle line, having folded the sides of the flanks and divided the sum of the money: m = (c + d)/2.
2. At the ACME trapezoid, described by the number of the stake, the sum of the dozhins of the bases is the sum of the dozhins of the side sides: AK + ME = KM + AE.
3. From the center of power, the trapezium is blatantly hardened: as many as you can fit into that trapezium, the sum of the sum of the costly sides.
4. The point of turning a stake with a radius r, inscribed in a trapezoid, splits the side of the edge into two sections, called їх a and b. The radius of the stake can be calculated using the following formula: r = √ab.
5. And one more power. Sob not to get lost, cross the butt yourself. We have the good old Acme trapezoid, the white of the stake is described. They have diagonals that overlap at the point O. The cutouts of the diagonals and the side sides of the AOK and EOM tricots are straight.
   The heights of these tricutniks, lowered on the hypotenuse (these are the side sides of the trapezium), are scaled with the radii of the inscribed stake. And the height of the trapezium - zbіgaєtsya with the diameter of the inscribed stake.

Dominance of a rectangular trapezium

A straight cut is called a trapezoid, one of the cuts is straight. І її authorities squeal from the surroundings.

1. In a rectangular trapezoid, one of the side sides is perpendicular to the bases.
2. The height is that side of the trapezium, which lies down to a straight kut, even. Tse allow you to calculate the area of ​​a rectangular trapezium (the capital formula S = (a + b) * h/2) not only through the height, but through the side, which lies down to a straight kut.
3. For a rectangular trapezoid, more relevant descriptions are given of the more powerful power of the trapezoid diagonals.

Prove the power of the trapeze

Rivnіst kuіv on pіdstavі rіvnofemoral trapezії:

• You may have already guessed for yourself that here we need a new trapezium ACME - to place an equal-femoral trapezium. Draw a straight line MT from the vertex M, parallel to the lateral side of AK (MT || AK).

Otrimany chotirikutnik AKMT - parallelogram (AK | | MT, KM | | AT). Oskіlki ME \u003d KA \u003d MT, ∆ MTE - rіnofemoral and MET \u003d MTE.

AK || MT, also MTE \u003d KAЄ, MET \u003d MTE \u003d KAЄ.

Stars AKM = 180 0 - MET = 180 0 - KAЄ = KME.

What did it take to bring.

Now, on the basis of the power of the equal-femoral trapezium (evenness of the diagonals), we can say that trapezium ACME є rіvnofemoral:

• On the back of the hand we can draw straight MX - MX || KE. Take away the parallelogram KMHE (substava - MX || KE i KM || EX).

∆AMH - equal thighs, scallops AM = KE = MX, and MAX = MEA.

MX || KE, KEA = MXE, to that MAE = MXE.

We have seen that the tricutniks AKE and EMA are equal to each other, to that AM = KE and AE are the main side of the two tricutniks. And also MAЄ \u003d MHE. We can make a non-trapeze visnovka, that AK = ME, and the stars are vibrating and that the trapezium AKME is isosceles.

Request for repetition

Substitute the trapezium AKME 9 cm and 21 cm, the side of the KA, which is 8 cm thick, makes the cut 150 0 with a smaller base. It is necessary to know the area of ​​the trapezium.

Decision: From the summit To the lowering height to the greater base of the trapezium. I need to look at the cut of the trapeze.

Kuti AEM and KAN are unilateral. And tse means that in sum stinks give 180 0 . To that KAN = 300 (on the basis of the quality of the cut of the trapezium).

Now let's look at the straight-cut ∆ANK (I know that this moment is obvious to readers without additional evidence). From the new we know the height of the trapezium KN - at the tricoutnik out with a leg, which lies opposite 30 0. To that KN \u003d AB \u003d 4 cm.

The area of ​​the trapezium is known by the formula: S AKME \u003d (KM + AE) * KN / 2 \u003d (9 + 21) * 4/2 \u003d 60 cm 2.

Pislyamova

So you respectfully and thoughtfully wove this article, did not use the olive in your hands to lay a trapezium for all the instigations of authority and to sort them out in practice, the material is guilty of being indiscreetly taken over by you.

Obviously, the information here is rich, varied and confusing: it’s not so easy to confuse the power described by the trapezoid with the power inscribed. Ale, you yourself have crossed over, what a difference is majestic.

Now you have a summary of the reports of all the great authorities of the trapeze. And also specific authorities and a sign of a trapezium equal-femoral and rectangular. I’m already handily currying, to get ready for the control tests and drinks. Try it yourself and share with friends!

site, with a full or private copy of the material sent to the original obov'yazkove.

The trapezium is the whole vapadok of a chotirikutnik, which has one pair of sides and is parallel. The term "trapezium" resembles the Greek word τράπεζα, which means "style", "table". We can see the trapezium and її dominion in these stats. In addition, let’s figure it out, how to build up the surrounding elements of the center, for example, the diagonal of the equal trapezium, the middle line, the square and the other. The material of contributions in the style of elementary popular geometry, that is, in an easily accessible form.

Zagalni vіdomostі

On the back, let's figure out what a chotirikutnik is like. Tsya figure є we will adorn the horn of the bagatokutnik, to avenge both the sides and the top. Two peaks of the chotirikutnik, yakі є susіdnіmi, are called opposite. The same can be said about the two inseparable sides. The main types of chotirikutniks are a parallelogram, a rectangle, a rhombus, a square, a trapezium and a deltoid.

Otzhe, let's turn to the trapezium. As we have already said, the position has two sides that are parallel. They are called fundamentals. Two sides (non-parallel) - sides. In the materials of experiments and other control robots, it is often possible to create a pattern, tied with trapeziums, which most often use knowledge that is not transferred by the program. Learn about the power of the cuts and diagonals, as well as the middle line of the equal-femoral trapezium. Adzhe, krіm tsgogo, a geometric figure has been created with a lot of other features. Ale about them a trifle zgod ...

See the trapeze

Іsnuє richly vidіv tsієї postati. However, it is most common to look at two of them - the straight-femoral one and the straight-cut one.

1. Rectangular trapezium - the whole figure, in which one of the lateral sides is perpendicular to the bases. She has two winds and ninety degrees.

2. Rivnofemoral trapezium is a geometrical figure, which has sides equal to each other. Otzhe, and kuti bіlya basics are also equal in pairs.

The main principles of the technique of raising the power of the trapezium

Before the main principle, it is possible to zarahuvat the so-called task approach. As a matter of fact, there is no need to introduce into the theoretical course of geometry new powers of figure. You can change and formulate in the process the variance of different tasks (more than systemic ones). In case of any other, it is important, knowing the vikladach, how it is necessary to put before the schoolchildren at that time the first moment of the initial process. Moreover, the skin power of the trapezium can be presented as a key task in the task system.

Another principle is the so-called spiral organization of the “wonderful” powers of the trapeze. Tse transferring the process of starting up to the last sign of this geometric position. In this rank, it is easier for me to learn to remember them. For example, the power of some point. Yogo can be brought up as if from the vivchennі similarity, and with the help of vectors. And the evenness of the tricots, lying to the side sides of the figure, can be brought, zastosovuyuchi as the power of the tricots with equal heights, drawn to the sides, like to lie one straight line, and with the additional formula S = 1/2 (ab * sinα). In addition, you can use it on an inscribed trapezium or a straight-cut tricot on a description trapezium, etc.

Zastosuvannya "behind the program" features of the geometric figure in the course of the school - the whole task of technology їх vikladannya. Steady stalking to the authorities, which rises when passing through other topics, allows the teachings to better recognize the trapezium and ensure the success of the accomplishment of the set goals. Otzhe, let's get down to the victory of the miraculous post.

Elements of power of the equal-femoral trapezium

As we have already designated, the sides of the geometric figure are equal. Another way out is how the trapeze is correct. And why is she so remarkable and why did she take such a name? To the peculiarities of the tsієї position lie those who have equals like a bіchnі side and kuti bіla basics, and th diagonals. In addition, the sum of the cuts of the equal-femoral trapezium is 360 degrees. And yet not all! With the help of the trapezium, it is hardly possible to describe the colo. Why is it connected with this, that the sum of the protilazhnyh kutіv tsієї figuri dori vnyuє 180 degrees, and for such a mind you can describe the column just about chotirikutnik. The advancing power of the analyzed geometric figure is those that rise from the top of the base to the projection of the adjacent top on a straight line, in order to avenge this base, it will be a healthy middle line.

And now let's figure out how to know the cuti of the equal-femoral trapezium. Let's take a look at the variant of the solution of the problem for the mind, that we can see the sides of the figure.

Solution

Zzvichay chotirikutnik is accepted to be denoted by the letters A, B, C, D, de BS and AT - ce substavi. The rіvnofemoral trapezії side of the rіvnі. Please note that їх rozmіr dorivnyuє X, and rozmіri pіdstav equal Y and Z (smaller and greater vіdpovіdno). To carry out the calculation, it is necessary to draw a height H from the kuta. Calculate the size of the leg AN: in the larger base, it is taken less, and the result is divisible by 2. Let's write down the formula: (Z-Y) / 2 = F. Now, for the calculation of the hot cut of the tricot, the cos function is faster. Let's take the next notation: cos(β) = Х/F. Now the cut is calculated: β=arcos (Х/F). Dali, knowing one cut, we can designate another, for which we perform an elementary arithmetic diu: 180 - β. Mustache kuti appointed.

Іsnuє and other vyshennya tsієї tasks. The back is lowered from the kuta At the height of N. The value of the BN leg is calculated. We know that the square of the hypotenuse of a rectangular tricot is equal to the sum of the squares of the catheters. We take: BN = √ (Х2-F2). We gave a victorious trigonometric function tg. As a result, we can: β = arctg (BN / F). Gostriy kut found. Dalі vyznaєmo similarly to the first method.

Authority of the diagonals of the equifemoral trapezium

We write down the chotiri rules on the back. If the diagonals in the equal-femoral trapezium are perpendicular, then:

The height of the figure is more expensive than the sum of the supports, subdivided into two;

Її height and middle line of rivers;

The center of the stake is a dot;

Yakshcho the side is divided by the point of torsional winding H and M, which is equal to the square root of the dobutka of these windings;

Chotiryokhkutnik, as if having become dotted with torsional points, the top of the trapezium and the center of the inscribed stake is a square, which side has a longer radius;

The area of ​​the post is more expensive, and the dobutku is laid down on the її height.

Similar trapezes

Tsya theme is already zruchna for the exercise of power tsієї. For example, the diagonals break the trapezium into chotiri tricoutniks, and they lie down to the bases - similar, and to the side sides - evenly large. Tse firmness can be called the power of trikutniks, where the trapezium is broken with diagonals. The first part of this hardness is brought through the sign of similarity for two kuts. To prove the other part, it is better to speed it up in a way that we point lower.

Theorem proof

Let's assume that the figure ABSD (AT and BS - the bases of the trapezium) is divided by the diagonals VD and AC. The point of the crossbar is O. Chotiri tricots are taken away: AOS - at the lower base, BOS - at the upper base, ABO and SOD at the side sides. Tricksters of the SOD and BOS sing their height in that fall, as if they were BO and OD є їх supports. It is important that the difference between their areas (P) is equal to the difference between them: PBOS / PSOD = BO / OD = K. Also, PSOD = PBOS / K. Similarly, BOS and AOB tricouters make a high-pitched height. Accepted for їх submissions of SB and OA. We take PBOS / PAOB \u003d CO / OA \u003d K and PAOB \u003d PBOS / K. Why do you swear that PSOD = PAOB.

To consolidate the material, it is recommended that students know the connection between the squares of the tricots, where the trapezium is broken with diagonals, violating such a task. Apparently, the area of ​​the trapezium is necessary to know the area of ​​the trapezium for tricots BOS and AOD. Oskіlki PSOD = PAOB, also, PABSD = PBOS + PAOD + 2 * PSOD. Z similar trikutnikiv BOS and AOD vyplivaє, scho BO / OD \u003d √ (PBOS / PAOD). Also, PBOS/PSOD = BO/OD = √(PBOS/PAOD). Let's take PSOD = √ (PBOS * PAOD). Todi PABSD \u003d PBOS + PAOD + 2 * √ (PBOS * PAOD) \u003d (√ PBOS + √ PAOD) 2.

Power like

Continuing the development of this topic, you can bring other features of the trapezium. So, for additional similarity, one can bring the power of a cross, which is to pass through a point, covered with a peretina of the diagonals of a geometric figure, in parallel to the basics. For this rozv'yazhemo zavdannya: it is necessary to know the length of the RK, which is to pass through the point O. Z similar to trikutnikov AOD and BOS vyplivaє, AO / OS = AD / BS. Z similar trikutnikiv AOR and ASB vyplivaє, shcho AO / AS = RO / BS = AD / (BS + AD). It is essential that RV = BS * AT / (BS + AT). Similarly, similarly to tricots DOK and DBS, it is obvious that OK \u003d BS * AD / (BS + AD). It is essential that RO=OK and RK=2*BS*BP/(BS+BP). The cross, which passes through the point of the crossbar of the diagonals, parallel to the bases and gets two sides, is divided by the point of the crossbar of the diagonals. Yogo dozhina - the middle harmonic of the figures.

Let's look at such a trapezoid, as if to name the power of some points. The points of the line of the diagonals (O), the line of the continuation of the lateral sides (E), as well as the middle of the bases (T and W) must always lie on the same line. It is easy to bring it up by the similarity method. Otrimanі tricoutniks BES and AED are similar, and in the skin їх median ET and ЄЖ the cut is cut at the top of the E equal. Also, points E, T and W lie on the same straight line. So it is on one straight line that the points T, O, and Zh are drawn. Zvіdsi robimo visnovok, scho all points - E, T, O and Z - lie on one straight line.

Vykoristovuyuchi such a trapezium, you can teach the learners to know the dozhina vіdrіzka (LF), which breaks the figure into two similar ones. Dany vіdrіzok is guilty but parallel to the basics. Oskіlki otrimani trapezії ALFD і LBSF podіbnі, BS/LF=LF/BP. Sounds obvious that LF=√(BS*BP). Let's take it apart, that it breaks the trapezoid into two similar ones, maє dozhina, equal to the average geometric dozhina of the foundations of the figure.

Let's take a look at such power similarities. Yoga is based on a vіdrіzok, which podіlyaє trapezі on tіvі іvnovіlіkі postаі. Let's assume that the ABSD trapezium is divided into two similar ones by the EP. From the top of B, the height is omitted, as if it is broken by a broken ЄP into two parts - B1 and B2. It is possible: PABSD / 2 \u003d (BS + EH) * B1 / 2 \u003d (AD + EH) * B2 / 2 and PABSD \u003d (BS + AD) * (B1 + B2) / 2. We then add up the system, first of all (BS + EH) * B1 \u003d (AD + EH) * B2 and the other (BS + EH) * B1 \u003d (BS + HELL) * (B1 + B2) / 2. It shows that B2 / B1 = (BS + EH) / (AD + EH) and BS + EH = ((BS + AD) / 2) * (1 + B2 / B1). It is important to note that the length of the trapezium is equal, that the trapezoid is divided into two equally large ones, to reach the mean quadratic length of the bases: √ ((BS2 + AD2) / 2).

Visnovki similarity

In this rank, we were brought, scho:

1. Vіdrіzok, shcho zadnuє at the middle of the trapezoidal side, parallel to AT and BS and to the arithmetic mean BS and AT (dovzhina of the base of the trapezії).

2. Rice, yak to pass through the point About the crossbar of the diagonals parallel to AT and BS, to reach the mean harmonic number AT and BS (2*BS*AD/(BS+AD)).

3. Vіdrіzok, scho breaking the trapezіy on podіbnі, maє dovzhina middle geometric bases BS and AT.

4. An element that divides the figure into two equally large ones, making the difference between the mean square numbers AT and BS.

In order to fix the material and to make sure the connection between the looked-out windows, I will learn how to call them for a specific trapezium. You can easily visualize the middle line and the top, which pass through the point O - the line of the diagonals of the figure - parallel to the bases. And the axis de will be rebuvat third and fourth? Tsya vіdpovіd led uchnya to vydkrittya shukanogo zv'yazku between average values.

Vіdrіzok, scho spoluchє middle of the diagonals of the trapezium

Let's look at such power of figure. Let's assume that the MN is parallel to the bases and subdivided the diagonals navpil. The points of the line are called W and Shch. Let's take a closer look. MSH - the middle line of the tricot ABS, out of dorіvnyuє BS / 2. MSC - the middle line of the tricot of the ABD, out of line AT / 2. It is also necessary that ShShch = MShch-MSh, also Shshch = AD / 2-BS / 2 = (AD + VS) / 2.

Vaga Center

Let's take a look at how the element is designated for this geometric figure. For which it is necessary to continue the foundations at the opposite side. What does it mean? It is necessary to add the lower base to the upper base - be it on the side, for example, to the right. And the lower one is moved to the upper left. Dali back to their diagonal. The point of the crossbar of the thogo vіdrіzka from the middle line of the figure and є the center of heaviness of the trapezії.

Inscribed and described trapezium

Let's review the features of such figures:

1. The trapezium can be inscribed in the bells only in that fall, as if it were even-femured.

2. If you can describe the trapezium, for you know, what is the sum of the dozhins of their subdavnyu the sum of the dovzhins of the side sides.

Recorded Stake Traits:

1. The height of the described trapezium is equal to two radii.

2. The flank side of the described trapezium is projected from the center of the stake under a straight cut.

The first consequence is obvious, but the proof of the other one needs to be established, which SOD is direct, which, in fact, is also not a warehouse of great practice. Then, knowing this power, allow, when solving tasks, to stop a straight-cut tricutnik.

Now we concretize the number of traces for the equal-femoral trapezium, as it is inscribed at the stake. We take into account that the height is the average geometrical support of the figure: H=2R=√(BS*AD). Vіdpratsovouchi main method of rozvyazannya zavdannya for trapezium (principle of holding two heights), the learner can virіshiti taka zavdannya. Let's assume that BT is the height of the equal-femoral figure of ABSD. It is necessary to know the details of AT and TD. Zastosovuyuchi formula, described above, it is not coherent.

Now let's figure out how to calculate the radius of the stake, the vicarious area of ​​the described trapezium. We lower the height from vertex B to the base AT. Oskіlki kolo is inscribed in a trapezium, then BS + AD \u003d 2AB or AB \u003d (BS + AD) / 2. From trikutnik ABN we know sinα = BN / AB = 2 * BN / (BS + AT). PABSD \u003d (BS + AT) * BN / 2, BN \u003d 2R. We take PABSD \u003d (BS + AD) * R, the reason is that R \u003d PABSD / (BS + AD).

Usі formulas of the middle line of the trapezium

Now the time has come to move on to the remaining element of this geometric figure. Let's figure out why the middle line of the trapezium is better (M):

1. Through the representation: M = (A + B) / 2.

2. Through the height, the basis of that kuti:

M \u003d A-H * (ctgα + ctgβ) / 2;

M \u003d B + H * (ctgα + ctgβ) / 2.

3. Through height, diagonals and cut between them. For example, D1 and D2 - diagonal trapezium; α, β - cut between them:

M = D1 * D2 * sinα / 2H = D1 * D2 * sinβ / 2H.

4. Through the area and height: M = P / N.

Trapeze- tse chotirikutnik, that there are two parallel sides, that are the bases of those two non-parallel sides, that are the side sides.

So zustrіchayutsya so called, like rіvnoboka or equal.

- The whole trapezium, near the kuti on the flank side is straight.

Trapeze elements

a, b- bases of a trapezoid(a parallel to b),

m, n- side sides trapezium,

d1, d2 diagonals trapezium,

h- elevation trapezії (vіdrіzok, scho z'єdnuє bases і with tsomu perpendicular їм),

MN- middle line(Vіdrіzok, scho behind the middle of the side walls).

Trapeze area

1. Through the sum of bases a, b and height h : S = \ frac (a + b) (2) \ cdot h
2. Through the middle line MN and height h: S = MN\cdot h
3. Through the diagonals d 1 , d 2 і cut (\sin \ varphi) between them: S = \frac(d_(1) d_(2) \sin \varphi)(2)

Domineering trapezium

Middle line of the trapezium

middle line parallel to the basics, dorovnyuє їх napіvsumі i podіlyaє kozhen vіdrіzok z kіntsami, scho to be on the straight lines, like avenging the foundations, (for example, the height of the figure) navpіl:

MN || a, MN | b, MN = \frac(a + b)(2)

Suma kutіv trapezії

Suma kutіv trapezії, which lies up to the skin side, 180 ^ (\circ) :

\alpha + \beta = 180^(\circ)

\gamma + \delta =180^(\circ)

Equally large tricot trapezium

Equally great, so that they can make equal areas, є ribs of diagonals and tricots AOB and DOC, made by side sides.

The similarity of the made tricots in the trapezium

Pull up the tricksє AOD and COB, as if they were embodied in their bases and in the arbors of the diagonals.

\triangle AOD \sim \triangle COB

Similarity coefficient k is behind the formula:

k = \frac(AD)(BC)

Moreover, the area of ​​\u200b\u200btheir tricots is increased to k ^ (2) .

Vіdnoshennya dovzhin vіdrіzkіv i pіdstav

Leather cross, which is the bottom of the foundation to pass through the point of the crossbar of the diagonals of the trapezium, subdivisions with the point of the line:

\frac(OX)(OY) = \frac(BC)(AD)

Tse will be fair and for the height of the diagonals themselves.