The diagonals of a rectangular trapezium are mutually perpendicular to the cut of the middle. Diagonal trapezium

1. Vіdrіzok, scho behind the middle of the diagonals of the trapezium, the older half of the retail pіdstav
2. Tricks, adorned with the bases of the trapezium and the ribs of the diagonals to the point of their crossbar - similar
3. Tricks, made with trapezoid diagonals, the sides of which lie on the side sides of the trapezium are equal in size (mayut the same area)
4. If you continue the side sides of the trapezium at the side of the smaller base, then the stench will cross over at one point from the straight line, which is the middle of the bases
5. Vіdrіzok, scho zadnuє bases of the trapezium, and pass through the point of the crossbar of the diagonals of the trapezium, share this point in proportion, so that the sp_vv_dnozhennju dozhina trapezii bases
6. Cross, parallel to the bases of the trapezium, and passing through the point of the crossbar of the diagonals, divide by the point navpil, and yogo dozhina is 2ab / (a ​​+ b), de a and b - the bases of the trapezium

The dominance of the cross, which happens to the middle of the diagonals of the trapezium

Therefore, the middle of the diagonals of the trapezoid ABCD, the result will have an LM triangle.
Vіdrіzok, scho behind the middle of the diagonals of the trapezium, lie on the middle line of the trapezium.

Denmark Vіdrіzok parallel to the bases of the trapezium.

Dovzhina vіdrіzka, shko z'єdnuє middle of the diagonals of the trapezium, dorіvnyuє vіvіrіznosti її osnovy.

LM = (AD-BC)/2
or
LM = (a-b)/2

The dominance of tricutniks, adorned with diagonals of a trapezoid

Tricots, as if made by the bases of the trapezium and the point of the crossbar of the diagonals of the trapezium - є similar.
Tricots BOC and AOD are similar. Shards of kuti BOC and AOD are vertical - the stench is equal.
Kuti OCB і OAD є vnutr_shnіmi lie side by side with parallel straight lines AD і BC (the bases of the trapezium are parallel to each other) and straight line AC, also, they stink equal.
Kuti OBC and ODA are equal to those reasons (internal lying sideways).

The oskіlki all three kuti of one tricoutnik are equal to the similar kutas of another tricoutnik, then these tricoutniks are similar.

What are you yelling at?

For the solution of problems in geometry, the similarity of trikutnikov vikoristovuetsya so. As we know the value of two important elements of such knitwear, then we know the coefficient of similarity (divided one by one). Zvіdki dovzhini reshti elementіv spіvvіdnosya among themselves with such very meanings.

The dominance of tricutniks, which lie on the side of the side and the diagonals of the trapezium

Let's look at two tricots, which lie on the side sides of the trapezium AB and CD. Tse - tricutniks AOB and COD. Irrespective of those who are different from the other sides of these trikutniks, they can be different, but squares of tricutniks, tucked by the side sides and the point of the crossbar of the diagonals of the trapezium rіvnі so trikutniks are equally large.

If you want to continue the sides of the trapezoid at the smaller base, then the point of the crossover of the sides will be zbіgatisya with a straight line, yak to pass through the middle of the bases.

In this rank, be it a trapezium, but it can be obtained to a trikutnik. With whom:

• Tricots, made with the bases of the trapezium from the crowned top at the point of the peretina of the prodovzhennyh sides, are similar
• The straight line that hits the middle of the foundations of the trapezium, is the median of the tricutnik

The dominance of the vіdrіzka, scho z'ednuє foundations of the trapezіy

If you want to hold a cross, which is to lie on the supports of the trapezium, which is to lie on the point of the crossbar of the diagonals of the trapezium (KN), then the spіvv_dnoshennia of the warehouse yoga vіdrіzkіv in the direction of the base to the point of the crossbar of the diagonals (KO / ON) bude spivvіdnjuє foundations of trapezії(BC/AD).

KO/ON = BC/AD

Tsya power is exuberant from the vіdpovіdny vіdpovіdnih trikutnikіv (wonderful vishche).

The dominance of the vіrіzka, parallel to the foundations of the trapezії

If you want to draw a cross, parallel to the bases of the trapezium and pass through the point of the crossbar of the diagonals of the trapezium, then in matima the advancing power:

• Vіdrіzok orders (KM) divide by a point the crossbar of the diagonals of the trapezium navpil
• Dovzhina vіdrіzka, to pass through the point of the crossbar of the diagonals of the trapezium and parallel to the bases, more KM = 2ab/(a + b)

Formulas for the meaning of the diagonals of a trapezoid

a, b- bases of a trapezium

c, d- side sides of the trapezium

d1 d2- diagonal trapezium

α β - kuti with a larger base of the trapezium

Formulas for the significance of the diagonals of a trapezoid through the bases, the side sides of that kuti at the base

The first group of formulas (1-3) reflects one of the main powers of the trapezoid diagonals:

1. The sum of the squares of the diagonals of the trapezium is more expensive than the sum of the squares of the side sides, plus the undercuts of the bases. The power of the diagonals of the trapezoid can be brought to the fore as a theorem

2 . This formula was taken away by the way of the transformation of the forward formula. The square of the other diagonal is thrown over the sign of equality, after which the square root is drawn from the left and right parts.

3 . The formula for the significance of the length of the diagonal of the trapezium is similar to that of the front, with the same margin, which in the left part of the viraz lacks the other diagonal

The advancing group of formulas (4-5) is analogous to change and turns analogously to spivvіdnoshennia.

A group of formulas (6-7) allows you to know the diagonal of the trapezoid, as you can see the larger base of the trapezoid, one side of the trapezoid at the base.

Formulas for the significance of the diagonals of a trapezoid in terms of height

Note. At this lesson, a solution of tasks from geometry about trapezium has been introduced. If you don't know the top problem of geometry, what to type you - ask the question on the forum.

manager.
The diagonals of the trapezium ABCD (AD | | BC) are tinted at point O. Find the base length of the BC base of the trapezoid, for example, the base AD = 24 cm, the length AB = 9 cm, the length OS = 6 cm.

Solution.
The development of this task of ideology is absolutely identical to the previous tasks.

Tricots AOD and BOC є similar to three kutіv - AOD і BOC є vertical, іnshі kuti in pairs equal, shards made by peretina of one line and two parallel lines.

Oskіlki trikutniks are similar, all їх geometrical dimensions are placed among themselves, as geometrical dimensions are given to us for the mind task of AO and OC. Tobto

AO/OC = AD/BC
9/6 = 24/BC
BC=24*6/9=16

Vidpovid: 16 cm

Manager.
The trapezium ABCD has AD=24, BC=8, AC=13, BD=5√17. Know the area of ​​the trapezium.

Solution.
For the significance of the height of the trapezoid from the vertices of the smaller base B and C, we drop it to the larger base of the two heights. Oskіlki trapeziya nerіvnoboka - significantly AM = a, KD = b ( do not confuse with the meanings of the formula znakhodzhennya ploschі trapezії). The shards of the base of the trapezoid are parallel, and we lowered two heights, perpendicular to the larger base, then MBCK is a rectangle.

To mean
AD=AM+BC+KD
a + 8 + b = 24
a = 16 - b

Tricots DBM and ACK are straight-cut, to that their straight cuts are made with trapezoidal heights. The height of the trapezium is significant through h. Follow the Pythagorean theorem

H 2 + (24 - a) 2 \u003d (5√17) 2
і
h 2 + (24 - b) 2 \u003d 13 2

It’s crazy that a \u003d 16 - b is the same in the first equal
h 2 + (24 - 16 + b) 2 \u003d 425
h 2 \u003d 425 - (8 + b) 2

We can imagine the value of the square of height from another equal, taken from the Pythagorean Theorem. We take:
425 - (8 + b) 2 + (24 - b) 2 = 169
-(64 + 16b + b) 2 + (24 - b) 2 = -256
-64 - 16b - b 2 + 576 - 48b + b 2 = -256
-64b = -768
b = 12

In this rank, KD = 12
Stars
h 2 \u003d 425 - (8 + b) 2 \u003d 425 - (8 + 12) 2 \u003d 25
h = 5

We know the area of ​​the trapezium through її height and pіvsumu pіdstav
, de a b - bases of the trapezium, h - height of the trapezium
S \u003d (24 + 8) * 5/2 \u003d 80 cm 2

Vidpovid: the area of ​​the trapezium is 80 cm 2

Znovu Pіfagorov trikutnik :))) As the pieces of the great diagonal in the great base to the point of the crosspiece are significant x, then obviously similar to the straight-cut trikutnikіv with the same kuta sld.х/64 = 36/х, zvіdsi x = 48; 4, to that, ALL rectilinear tricots, made with bases, diagonals and the side side, perpendicular to the base, are similar to the tricot with sides 3,4,5. Vinyatok to become less trikutnik, tucks with shmatkas of diagonals and an oblique side, ale vines are not for us:). (If it was sensible, similar, about how to go - it’s less than NAMED BY ІNSHOM trigonometric functions of cutiv:) we already know the tangent of kuta between the great diagonal and the great base, vin dorivnyu 3/4, mean sine of dorіvnyu, є 3 cosine 4/5 :)) You can write to me

Vidpovidi. The lower base 80 height of the trapezium will be 60, and the upper one - 45. (36*5/4 = 45, 64*5/4 = 80, 100*3/5 = 60)

Similar tasks:

1. The base of the prism is a tricutnik, in which one side is 2 cm long, and two sides are 3 cm each.

2. Supporting a frail prism є equilateral tricot with side a; one of the side faces is perpendicular to the plane of the base and is a rhombus, for which the diagonal is longer. Find out about the prism.

3. In a frail prism, the base is a straight-cut tricout, the hypotenuse of which is dear, one host kut 30, the bichne rib is darn to and folds over the flat base of kut 60. Know the volume of the prism.

1. Find the side of the square, so that the diagonal becomes 10 cm

2. At the equal-femoral trapezium, the blunt cut is 135 degrees less than the base of the trapezium is 4 cm, and the height is 2 cm to know the area of ​​the trapezium?

3. The height of the trapezium is 3 times greater for one of the substations, and two times less for the second. Know the basics of the trapezium and the height of the area of ​​the trapezium is 168 cm near the square?

4. In tricotnik ABC kut A \u003d B kuti \u003d 75 degrees. Know the sun, as if the square of the tricot is 36 cm near the square.

1. At the trapezium ABCD with side sides AB and CD, the diagonals intersect at point O

a) Adjust the area of ​​tricots ABD and ACD

b) Adjust the area of ​​tricots ABO and CDO

c) Tell that OA*OB=OC*OD

2. The base of the equal-femoral tricot is extended to the side of the yak 4:3, and the height, drawn to the base, is 30 cm.

3. The line is AM-dotted to a stake, and AB is a chord of that stake. To add that the MAB kut is reduced by the half of the AB arc, ruffled in the middle of the MAB kut.

As in the equal-femoral trapezium, the diagonals are perpendicular, in the case of the highest task, the theoretical material will be inverted.

1. As in the equal-femoral trapezium, the diagonals are perpendicular, the height of the trapezium is more beautiful than the basics.

Draw line CF through point C, parallel to BD and continue line AD to the crossroads from CF.

Chotiriokhkutnik BCFD - parallelogram (BC DF as the basis of the trapezium, BD CF as a reminder). So CF=BD, DF=BC and AF=AD+BC.

The tricot ACF is rectangular (if a straight line is perpendicular to one of two parallel lines, then it is perpendicular to another straight line). Shards in a rіvnofemoral trapezії diagonally rіvnі, and CF \u003d BD, then CF \u003d AC, then the tricot ACF is rіnofemoral with the base AF. So, the height of CN is also the median. Shards of the median of the rectocut tricutnik, carried out to hypotension, halfway through, then

what can be written down by a savage spectator

de h - height of the trapezium, a and b - її bases.

2. Since the equal-femoral trapezium is diagonally perpendicular, then the height is longer than the middle line.

Shards of the middle line of the trapezium m dor_vnyu nap_vsum_ basics, then

3. As in the equal-femoral trapezium, the diagonals are perpendicular, then the area of ​​the trapezium is equal to the square of the height of the trapezium (either to the square of the nape of the bases or to the square of the middle line).

Shards of the area of ​​the trapezium are known for the formula

and the height, nap_vsuma of the base and the middle line of the equal trapezium with perpendicular diagonals of the equal between themselves:

4. As in the equal-femoral trapezium, the diagonals are perpendicular, then the square of the її diagonal to the half of the square of the sum of the bases, as well as to the double square of height and the double square of the middle line.

So, like the area of ​​a swollen chotirikutnik, you can know through the yogo diagonals and cut between them by the formula