Visota is drawn from the top of the straight kuta є. Rectangular tricycle. I. For the Gosting Kuta

Trikutniki.

Basic understanding.

trikotnik- the whole figure, which is stored in three ways and three points, which do not lie on the same straight line.

are called parties, And points - peaks.

suma kutiv tricycle road 180 º.

The visota of the tricot.

Tricot's visota- central perpendicular, conduction from the top to the opposite side.

At the gostrokutn the tricytnyk is located in the middle of the tricutnik (Fig. 1).

In the rectangular tricot, the legs are є with the visota of the tricut (Fig. 2).

In an obtuse-angled tricytube, the visota should pass the posture by a tricytube (Fig. 3).

The power of the tricycle:

Trikutnik bisektrix.

bisector tricutnik- a chain of ends, which extends the top of the summit navpil and from the bottom top from a point on the opposite side (Fig. 5).

Power bisectrix:

Mediana tricycle.

mediana tricycle- tse vіdrizok, scho z'єnu apex from the middle of the protolezhny side (Fig. 9).

Dovzhinu mediani can be calculated by the formula:

2b 2 + 2c 2 - a 2
m a 2 = ——————
4

de m a- mediana, carried out to the side a.

For the rectangular tricycle, the median was carried out before the hypotenuse, half of the hypotenuse:

c
m c = —
2

de m c- medіana, carried out before hypotenuse c(Fig.9c)

The median of the tricycle is intertwined in one point (in the center of the tricycle) and is distributed at the same point in the ratio 2: 1, in the direction of the peak. Tobto goes from the top to the center two times more from the center to the side of the trike (pic.9c).

Three median tricycles to carry on to a number of equally large tricycles.

Middle line of tricycle.

Middle line of tricycle- tse vіdrіzok, scho z'єdnu the middle of the two sides (Fig. 10).

The middle line of the tricycle is parallel to the third side and the road half

The name of the trikutnik kut.

zovnishniy kut trikutnik dorіvnyu sumi two incompetent internal kutiv (Fig. 11).

The famous kut of a tricycle is bigger than any crazy kut.

Straight-cut tricycle.

Straight-cut tricycle- tricut, for yakogo є straight kut (Fig. 12).

The side of the rectangular tricycle, opposite to the straight edge, is called hypotenuse.

The two sides are called legs.

Proportional drifts in a rectangular tricycle.

1) In the rectangular tricycle, the visota was carried out straight kuta, I will approve three separate tricycles: ABC, ACH and HCB (fig. 14). Apparently, kuti, approved by the height, equal kutah A and V.

fig. 14

Equestrian tricutnik.

Equestrian tricutnik- tricycle, at which two sides of the ryvn (Fig. 13).

ci equal parties call myself business parties, And the third - before trikutnik.

In tricycle kuti at the base of the pivot. (To our tricotnik kut A dorivnyu kut C).

At the trikutnik medіana, carried out to the base, є at once і bіsectrix, і by the height of the tricyаtnik.

Equilateral trikutnik.

Equilateral tricycle - tricytric, at which all sides are equal (Fig. 14).

The power of the equilateral trikutnik:

Miracles of power of trikutniks.

In tricycles, they are the original authorities, who can help you to successfully complete the work, knitted with tsimi figures. Acts of power Ale will repeat it again, adding to them a sprinkle of these wonderful features:

1) For a rectangular tricot with 90º, 30º and 60º legs b To lie on the edge of the kuta at 30 degrees, on the way half of hypotenuse. And the lega more legb√3 times (Fig. 15 a). For example, if the leg b is 5, then the hypotenuse c obov'yazkovo door 10, and the leg a door 5√3.

2) In a rectangular tricycle with cutouts 90º, 45º and 45º, hypotenuse is √2 times larger than the leg (Fig. 15 b). For example, if the catheti is equal to 5, then the hypothesis is 5√2.

3) The middle line of the tricycle to the half parallel to the side (Fig. 15 s). For example, if the side of the tricycle is road 10, then it is parallel to the middle of the road 5.

4) In the rectangular tricycle median, carried out before the hypotenuse, half of the hypotenuse (pic.9v): m c= C / 2.

5) The median of the tricycle, crossing in one point, is distributed by the central point in the relationship 2: 1. To go from the top to the point of crossing the median two times more than that of the point to the side of the median to

6) In the rectangular tricycle, the middle of the hypotenuzi є by the center of the described stake (Fig. 15 d).

Signs of equanimity of tricycles.

First sign of eagerness: If there are two sides and cut between them of one trike, then two sides and cut between them are the same trike, then there are triplets.

Another sign of eagerness: If the side and the cut of one tricycle are attached to it, then we will fit the sides to the cut of one tricot, then these are the tricycles.

The third sign of equanimity: If three sides of one tricycle are equal to the three sides of another tricycle, then such tricycles are equal.

Invalidity of the tricycle.

Any tricycle has a skin side that is less than two sides.

Pyfagor's theorem.

The rectangular tricycle has a square of hypotenuzi for the sum of squares of legs:

c 2 = a 2 + b 2 .

The area of the tricot.

1) The area of the tricycle for the second half of the work is done on the side to the top, drawn to the center of the side:

ah
S = ——
2

2) The area of the tricycle is two halves of the two sides on the sinus of the cut between them:

1
S = — AB AC · sin A
2

Trikutnik, descriptions close to the circle.

Colo is called inscribed in a trikutnik, if you won’t be able to touch all the sides (Fig. 16 a).

Trikutnik, fittings in the colo.

The trikutnik is called inscribed in the colo, as it is the fault of all the peaks (Fig. 17 a).

Sine, cosine, tangent, cotangent of the gost kut of a rectangular tricycle (Fig. 18).

sinus gostry kuta x protylezhny leg to hypotenuse.
Start like this: sinx.

cosine gostry kuta x rectangular tricycle - price of delivery snug leg to hypotenuse.
To mean like this: cos x.

tangent gostry kuta x- the price of the prototyping leg to the flat leg.
Start like this: tgx.

cotangent gostry kuta x- the price of a close leg to a prototype one.
Start like this: ctgx.

Rules:

Cathetus, prototype kutku x, Dobutku hypotenuzi on sin x:

b = c Sin x

Leg, lying down to the kut x, Dobutku hypotenuzi on cos x:

a = c Cos x

Cathetus, opposite kutu x, Dobutku add another leg to tg x:

b = a Tg x

Leg, lying down to the kut x, Dobutku add another leg to ctg x:

a = b Ctg x.

For be-like a gosty kut x:

sin (90 ° - x) = Cos x

cos (90 ° - x) = Sin x

Power: 1. In any kind of rectangular tricycle, the visota is lowered from the straight kuta (to the hypotenuse), to dilute the rectangular tricot, into three additional tricycles.

Power: 2. The height of the rectangular tricycle is lowered to the hypotenuse, to the middle geometric projection of the legs to the hypotenuse (or the middle geometric quiet to the hypotenuse).

Power: 3. The leg is for the middle geometric hypotenuse and the projection of the leg onto the hypotenuse.

Power: 4. The leg is opposed at 30 degrees to the side of half of the hypotenuse.

Formula 1.

Formula 2., De hypotenuse; , Kateti.

Power: 5. In the rectum tricycle, the median was carried out before the hypotenuse, the second half and the second half of the described stake.

Power: 6. Overlap between the sides and kutas of the rectangular tricycle:

44. Cosine theorem. Traces: links between diagonals and sides of a parallelogram; type of tricycle; the formula for calculating the amount of median tricycle; the calculation of the cosine of the tricot kuta.

Kinets of robots -

Qia theme should be distributed:

Klas. Coloquium program basics of planimetry

power sumy huts..that two kutas are summed up, because one side of them is in the back of the two, they set up a straight line ..

If you need additional material on this topic, for you didn’t know those who were joking, it is recommended to make a quick joke on our base of robots:

We will be robbed with material:

As soon as the material has appeared cinnamon for you, you can save it to your side in social framing:

Straight-cut tricycle- a tricycle, in which one of the cuts is straight, so that it travels to 90 degrees.

The side opposite to the straight kuta is called the hypotenuse (the yak c or AB)
The side that lies down to the straight kuta is called the leg. Kozhen rectangular trikutnik has two legs (a yak is designated for a baby a= b or AC = BC)

Formulas and power of a rectangular tricycle

Formulas value:

(Div. Figure vishche)

a, b- legs of a rectangular tricycle

c- hypotenuse

α, β - gostrі kuti trikutnik

S- area

h- hanging, lowered from the top of the straight kut to the hypotenuse

m a a from the protyezhny kut ( α )

m b- mediana, carried out to the side b from the protyezhny kut ( β )

m c- mediana, carried out to the side c from the protyezhny kut ( γ )

V rectangular tricycle be it with cathetes less hypotenuzi(Formulas 1 and 2). Power is given to the ancestral theorems of Pythagoras.

Cosine of be-like z gostrich kutiv less than one (Formulas 3 and 4). The power of the drink is given from the front. So, be it from the cathetes less than the hypotenuse, then from the spacing of the leg to the hypotenuse is less than one.

The square of hypotenuse for the sum of squares of legs (Pyphagoras' theorem). (Formula 5). Tse the power is constantly victorious when dealing with tasks.

Rectangular tricycle area create half of the legs (Formula 6)

Suma kvadіv medіan up to the legs, up to five squares of median to hypotenuse and five squares of hypotenuse, seen on chotiri (Formula 7). Крім is designated, є more 5 formulas Therefore, it is recommended to learn about it also with the lesson "Medicine of a rectangular tricycle", in which more detailed information is given to the power of media.

Visota rectangular tricycle with additional legs for hypotenuse (Formula 8)

The square of the legs is wrapped in the proportional square of the height, lowered to the hypotenuse (Formula 9). The same is given by one of the inherited theorems of Pythagoras.

dozhina gіpotenuzi to the diameter (two radii) of the described stake (Formula 10). Hypotenuse of rectangular tricycle є the diameter of the circumscribed circle... The power is often victorious when dealing with tasks.

radius of inscribed v rectangular tricycle circumference You can know about half of the viraz, which includes a bag of legs for a tricycle minus to the amount of hypotenuse. Anyway, like a pair of cathetes, the sum of all sides (perimeter) of a given trike is added to the bag. (Formula 11)
sinus kuta vіdnoshennyu protylezhny this rose leg to hypotenuse(For the sinus value). (Formula 12). Given the power to be victorious when assigning tasks. Knowing the magnitude of the sides, you can know the kut that smells like it.

Cosine kuta A (α, alpha) in the rectangular tricycle will be suitable vidnoshennyu snug this rose leg to hypotenuse(For the sinus value). (Formula 13)

(ABC)і th power, like representations for the little one. The rectangular tricycle can be hypotenuse - aside, lie on the opposite side of the straight kuta.

Rada 1: Yak to know the visota in the rectangular tricycle

The parties that make a straight kut are called legs. On the baby's side AD, DC and BD, DC- legs, and sides ASі SV- hypotenuse.

Theorem 1. In a rectangular tricycle with a cut of 30 ° legs, half of the hypotenuse breaks in the opposite cout.

AB- hypotenuse;

ADі D B

trikotnik
The basic theorem:
comment system CACKLE

Decision: 1) Diagonals of any straight-line plain. True 2) If there is one gostry kut in a tricut, then the whole tricut is a gostrokutny. Not true. See tricutniks. A trikutnik is called gorodkutny, as all three yogi kuta are gosti, tobto less than 90 ° 3) Where the point is to lie on.

Abo, in the іnshomu record,

By the theorem of Pyphagoras

Chomu dorіvnyuє visota in a rectangular tricycle formula

Visota of a rectangular tricycle

The height of the rectangular tricycle, carried out before the hypotenuse, can be found in the same way in the fallowness as given in the mind of the plant.

Abo, in the іnshomu record,

De BK and KC projections of the cathetes for the hypotenuse (for example, at the same height to extend the hypotenuse).

Visota, carried out to hypotenuse, can be known through the area of the rectangular tricycle. Yaksh zastosuvati formula for znakhozhennya area of tricycle

(Half of the work is done sideways to the top, carried out to the central side) before the hypotenuse and the top, carried out before the hypotenuse, we can say:

It is possible to know how high it is from the bottom of the tricycle area before the hypotenuzi:

So, like the area of a rectangular knit, create half of the legs:

Tobto a dose of power, carried out before hypotenuse, create cathetes before hypotenuse. How to denote by a and b, by c, the formula can be rewritten in a viglyad

So, as the radius of the stake described near the rectangular tricycle, half of the hypotenuse, for the first time, can be violated through the legs and the radius of the described stake:

The test was carried out up to the hypotenuse height, which was made up to two rectangular tricycles, and you can know it through the experience in the rectangular tricycle.

3 rectangular tricycle ABK

3 rectangular tricycle ACK

It is possible to rotate through the legs of a rectangular tricycle for more lengths of time. So yak

By the theorem of Pyphagoras

How to square the offense of the part of the ryvnosti:

It is possible to correct one formula for the ligament of a rectangular tricycle with legs:

Chomu dorіvnyuє visota in a rectangular tricycle formula

Rectangular tricycle. Middle Riven.

Do you want to reconsider your strength and see the result as much as ready to ADI or OGE?

The main theorem about the rectangular tricycle is Pyfagor's theorem.

Pyfagor's theorem

Before the speech, why are you kindly pam'yatash, why are you also kateti and hypotenuse? If it’s not even worse, then marvel at the little one - learn the knowledge

All in all, but also richly developed the vicorist theorem of Pythagoras, and the axis of thought was misleading, for which such a theorem is true. Yak bi її to bring? And come on brutally, like the ancient Greeks. Namaluєmo square with a side.

Bachish, as cunningly we poured yo side on the occasion of the dinner!

And now we have seen points

Here we, it is true, still have denoted, scarlet yourself captivated by the little ones and think, why so.

Why is there a big square? Right,. A smaller area? Zvisno,. The total area of chotir'okh kuchk_v has been overshadowed. See if they took them two at a time and put one to one hypotenuzi. What happened? Two rectangulars. This means that the area is "obr_zk_v" dorіvnyu.

Let's take everything at once.

Axis i was touched by Pythagoras - they brought his theorem in an ancient way.

Rectangular tricycle and trigonometry

For a straight-backed tricycle, check the offensive performance:

The sinus of the gostry kut is used to replace the protruding leg to the hypotenuse

The cosine of a good leg is from the position of an adjacent leg to a hypotenuse.

The tangent of the gosty kut is from the leading leg to the snug leg.

The cotangent of the gooseneck leg is brought from the adjacent leg to the protruding leg.

Once again, all the signs are at the viglyadі:

Why do I remember one more tightly? Subscribe to the sign respectfully.

Tse duzhe zruchno!

Signs of equanimity of upright trikutniks

II. For the leg and hypotenuse

III. For hypotenuse and gostroma kutu

IV. For a leg and a gostroma kuta

Uwaga! Here it is even more important, the legs of the legs are "vidpovidnі". For example, it will be like this:

Those TRIKUTNIKI NOT RIVNІ, Uninvolved on those who show one by one the same guest kuta.

Needed, schob In both tricytniks, the leg buv will fit, or in both - with the opposite.

And, having said, what are the signs of equanimity of upright triketers from the most common signs of equanimity of triketers? Zazirni in the theme of "Trikutnik" and animals I respect for those, but for the equanimity of "ordinary" trikutniks, three elements are needed: two sides and a cut between them, two kutas and a side between them or three sides. And the axis for the equilibrium of rectangular tricycles is to reach two different elements. Great, isn't it?

Approximately the same is the situation with signs of similarity of rectangular tricycles.

Signs of the characteristics of straight-backed tricycles

III. For a leg and hypotenuse

Mediana in rectangular tricycle

It is easy to see the replacement of the rectangular tricycle.

The diagonal is drawn and the point of the overflow of the diagonals is visible. What about the rectangular diagonals?

With a precise cross-flow of the diagonal to distribute to the diagonal to the diagonal

What is the best drink?

Axis i viishlo, scho

Remember the fact! Douzhe dopomagaє!

And it’s more wonderful, so it’s good, it’s true and sound.

What good can be said about the fact that the median was carried out before the hypotenuse, half of the hypotenuse? Let's marvel at the picture

Surprise respectfully. We have є :, so that the points from the point to all three peaks of the tricycle have come to the point. Ale in the tricycle є there is only one point, which is about all three peaks of the tricycle of the rivn, and it is the CENTER of the described stake. So, what happened?

Give me an axis for some reason. ".

Ale at the junior trikutikiv all kuti rіvnі!

The same can be said about i

And now all at once:

Have the same gostri kuti!

The yak, the cinnamon, can be otrimatized from the tsy "tricky" likeness.

Well for nothing - Two formulas for the height of the rectangular tricycle.

Recordable for each side:

For the knowledge of the visoti, the proportion is virishuєmo I persuade the formula "Visota in a rectangular tricycle":

Yak, right, to your friend?

And now it’s stagnant similar to trikutnikiv i.

Otzhe, you can zastosuvati butt :.

What will you do now?

I know the virile proportion and accept the formula to a friend "Visota in a rectangular tricycle":

The offense of formulas is required for even better memory and storage for that, which is better. Writable їх more times

Well, the axis, now, zasosovuchi and combinuyuchi and znannya zinshih, ty vyrishish be-yak zabadannya with a straight-headed tricycle!

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Power of the rectangular tricycle

Clearly straight-cut tricycle (ABC)і th power, like representations for the little one. The rectangular tricycle can be hypotenuse - aside, lie on the opposite side of the straight kuta. The parties that make a straight kut are called legs. On the baby's side AD, DC and BD, DC- legs, and sides ASі SV- hypotenuse.

Signs of equilibrium of a rectangular tricycle:

Theorem 1. If the hypotenuse and the legs of the rectangular tricycle are similar to the hypotenuse and the leg of the first tricycle, then the same tricytes are equal.

Theorem 2. If two legs of a straight-leg tricycle go to two legs of a tricycle tricycle, then such tricycles are pivot.

Theorem 3. If the hypotenuse and gostry kut of a rectangular tricycle is similar to a hypotenuse and a host kut of a rectangular tricycle, then such tricycles are equal.

Theorem 4. If the leg and the adjoining (prototypal) gostry kut of the rectangular tricycle, the leg and the adjoining (protylezhny) kut of the rectangular tricycle, then the tricycles of the rivn.

The power of the leg, opposite the cut at 30 °:

Theorem 1.

Visota in a rectangular tricycle

In a rectangular tricycle with a cuff of 30 °, half of the hypotenuse breaks in the protruding cout.

Theorem 2. If there is half of the hypotenuse in the rectangular tricycle, then the other half of the warehouse is 30 °.

If the height is carried out from the top of the straight cut to the hypotenuse, then such a tricycle extends into two smaller ones, from the top to the outward and analogous one to the last. Three types of visnovka are used:

Visota є middle geometric (middle proportional) two segments of hypotenuse.
The skin leg of the tricycle is in the middle proportional hypotenuse and summery segments.

At the rectangular tricycle, in the role of the heels, the legs are played. Orthocenter - tse such a point, on which the perets of the tricytes are seen. Vona zbіgaєtsya at the top of the straight kuta figuri.

hC- hang out from the straight kuta of the tricot;

AB- hypotenuse;

ADі D B- vіdrіzki, wіnіkli with rozpodіlі hypotenuse with visota.

Turn around to see the details of the discipline "Geometry"

trikotnik- tse geometric figure It is possible to store from three points (peaks), as it does not perebuyut on one and the same straight line and three points from the point. A straight-cut tricyt is called a tricut, which is one of the cuts at 90 ° (straight cut).
The basic theorem: bag gostrykh kutіv rectangular trikutnik dorіvnyuє 90 °.
comment system CACKLE

Key words: trikutnik, rectangular, leg, hypotenuse, Pythagorean theorem, circumference

call tricycle straight-headed, Yaksho u nyogo є straight kut.
The straight-cut tricycle can be doubled perpendicular to the sides, called legs; third party to be called hypotenuse.

Behind the power of the perpendicular and the abducted hypotenuse is longer than the cutaneous catheter (ale less than the sumi).
Suma two gostrich kut_v of a rectangular tricot for a dor_vnyuj of a straight kut.
Two legs of a rectangular tricycle are attached with legs. Tom, one of the chotir'okh of the miraculous points is at the top of the straight kuta tricut.
The center of the described stake of a rectangular tricycle should lie in the middle of the hypotenuzi.
The median of the rectangular tricycle, drawn from the top of the rectangular tricycle to the hypotenuse, є by the radius of the circumference of the circumference of the circumference.

A clear-cut rectangular tricycle ABC і is carried out at the height of CD = hc from the top of the C th straight kuta.

Vona rozib'є Daniy trikutnik for two rectangular trikatniks АСD and ВСD; kozen z tikh trikutnik_v maє z trikutnik ABC zagalny gostry kut і that is similar to the trikutnik ABC.

All three tricycles ABC, ACD and BCD are similar to each other.

Three similar tricycles on the visitor start with the following:

$$ h = \ sqrt (a_ (c) \ cdot b_ (c)) = \ frac (a \ cdot b) (c) $$;
c = ac + bc;
$$ a = \ sqrt (a_ (c) \ cdot c), b = \ sqrt (b_ (c) \ cdot c) $$;
$$ (\ frac (a) (b)) ^ (2) = \ frac (a_ (c)) (b_ (c)) $$.

Pyfagor's theorem one of the main theorems of Euclidean geometry, which will establish the relationship between the sides of the rectangular tricycle.

Geometric formula. The rectangular tricycle has an area of a square, prompted on hypotenuse, a large sum of areas of squares, prompted on legs.

Algebraic formula. The rectangular tricycle has a square of hypotenusi for the sum of squares of legs.
Tobto, having denoted some of the gipotenuzi of the tricycle through c, and the more of the legs through a and b:
a2 + b2 = c2

Zvorotn_y Pyfagor's theorem.

Visota of a rectangular tricycle

For be it a trio of positive numbers a, b and c, such,
a2 + b2 = c2,
іnuє rectangular tricycle with legs a і b і hypotenuse c.

Signs of equanimity of straight-backed tricycles:

on the leg і hypotenuse;
for two legs;
on leg and gostrom kut;
on hypotenuse and gostrom kut.

Div. also:
Square of a trikutnik, Equilateral trikutnik, Equilateral trikutnik

Geometry. 8 class. test 4. option 1 .

AD : CD = CD : BD. Zvidsey CD2 = AD ∙ BD. seem to be:

AD : AC = AC : AB. Stars AC2 = AB ∙ AD. seem to be:

BD : BC = BC : AB. Zvidsi BC2 = AB ∙ BD.

Visiting staff:

A) 70 cm; B) 55 cm; C) 65 cm; D) 45 cm; E) 53 cm.

2. The height of the rectangular tricycle, carried out before the hypotenuse, extend the hypotenuse to the lengths 9 and 36.

Visitor to the dinner of the price.

A) 22,5; B) 19; C) 9; D) 12; E) 18.

A) 30,25; B) 24,5; C) 18,45; D) 32; E) 32,25.

A) 25; B) 24; C) 27; D) 26; E) 21.

A) 8; B) 7; C) 6; D) 5; E) 4.

8. Leg of a rectangular tricycle leg 30.

Yak know the visota in the rectangular tricycle?

To know the distance from the top of the straight kut to the hypotenuse, which is the radius of the road described around the tricycle of the road 17.

A) 17; B) 16; C) 15; D) 14; E) 12.

10.

A) 15; B) 18; C) 20; D) 16; E) 12.

A) 80; B) 72; C) 64; D) 81; E) 75.

12.

A) 7,5; B) 8; C) 6,25; D) 8,5; E) 7.

See the messages!

D8.04.1. Proportional trips in a rectangular tricycle

Geometry. 8 class. test 4. option 1 .

At Δ АВС ∠АСВ = 90 °. АС and ВС catheti, AB hypotenuse.

CD height of the tricut, held before hypotenuse.

AD projection of the AC leg onto the hypotenuse,

BD projection of the BC leg onto the hypotenuse.

Vysota CD to divide the tricycle ABC into two additional ones (and one to one) tricycle: Δ ADC і Δ CDB.

Proportional sides of some Δ ADC and Δ CDB next:

AD : CD = CD : BD.

The power of the head of the rectangular tricycle, lowered to the hypotenuse.

Zvidsey CD2 = AD ∙ BD. seem to be: the height of the rectangular tricycle, carried out before the hypotenuse,є the average proportional value between the projections of the legs to the hypotenuse.

З similarities Δ ADC і Δ АCB next:

AD : AC = AC : AB. Stars AC2 = AB ∙ AD. seem to be: skin leg є average proportional size between all hypotenuse and projection of the leg onto the hypotenuse.

Similarly, with the option of Δ СDВ і Δ АCB followed:

BD : BC = BC : AB. Zvidsi BC2 = AB ∙ BD.

Visiting staff:

1. Know the height of the rectangular tricycle, carried out to the hypotenuse, if you want to extend the hypotenuse to the lengths of 25 cm and 81 cm.

A) 70 cm; B) 55 cm; C) 65 cm; D) 45 cm; E) 53 cm.

2. The height of the rectangular tricycle, carried out before the hypotenuse, extends the hypotenuse to the lengths of 9 and 36. Visot to the amount of the hypotenuse.

A) 22,5; B) 19; C) 9; D) 12; E) 18.

4. The height of the rectangular tricycle, carried out to the hypotenuse, door 22, the projection of one of the leg legs is 16. Know the projection of the second leg.

A) 30,25; B) 24,5; C) 18,45; D) 32; E) 32,25.

5. The leg of the rectangular tricycle is 18, and the projection onto the hypotenuse is 12. Know the hypothesis.

A) 25; B) 24; C) 27; D) 26; E) 21.

6. Household hypotenuse 32. To know the leg, the projection of which on the road hypotenuse 2.

A) 8; B) 7; C) 6; D) 5; E) 4.

7. Hypotenuse of a rectangular tricycle to the door 45. To know the leg, the projection of which to the hypotenuse of the door 9.

8. Leg of a rectangular tricycle road 30. To know the distance from the top of a straight tricycle to a hypotenuse, as the radius of the described near tricut tricycle road 17.

A) 17; B) 16; C) 15; D) 14; E) 12.

10. The hypotenuse of the rectangular tricycle is 41, and the projection of one of the legs is 16. Know the length of the leg, drawn from the top of the straight kut to the hypotenuse.

A) 15; B) 18; C) 20; D) 16; E) 12.

A) 80; B) 72; C) 64; D) 81; E) 75.

12. Rise of the projection of the cathetes to the hypotenuse of the road 15, and from the top of the straight kut to the hypotenuse of the road 4. Know the radius of the described stake.

A) 7,5; B) 8; C) 6,25; D) 8,5; E) 7.

As a matter of fact, all calls are not so scary. Obviously, the "reference" value of the sine, cosine, tangent and cotangent should be astonished in the statty. Ale doesn't even want to, does she? We can make you happy: for the news about the upright tricycle, you can simply memorize the next simple speech:

What about kut? Chi є cathetus, which is located on the other side of the kuta, what is the counterpart (for the kuta) cathetus? Wonderful є! Tse cathetus!

And the yak is the kut? Surprise respectfully. Yaky cathetus lie down to kuta? Well, cathetically. So, for the kuta, the legs are lying down, i

And now, uvaga! Submit it, we got it:

Bachish, yak great:

Now we go to tangent and cotangent.

How can you write it down in words now? Katet yakim є according to delivery to kuta? Protylezhny, zychano - vin "to lie" against the kuta. And the leg? Lie down to kuta. So, what happened to us?

Bachish, chiselnik and znamennik remembered in mots?

I now know the kuti and the exchange was made:

summary

Let's briefly write down everything that we know.

Pyfagor's theorem:

The main theorem about the rectangular tricycle is Pyfagor's theorem.

Pyfagor's theorem

Before the speech, why are you kindly pam'yatash, why are you also kateti and hypotenuse? If it’s not even worse, then marvel at the little one - learn the knowledge

Bachish, as cunningly we poured yo side on the occasion of the dinner!

And now we have seen points

Here we, it is true, still have denoted, scarlet yourself captivated by the little ones and think, why so.

Why is there a big square?

Right,.

A smaller area?

Zvisno,.

The total area of chotir'okh kuchk_v has been overshadowed. See if they took them two at a time and put one to one hypotenuzi.

What happened? Two rectangulars. This means that the area is "obr_zk_v" dorіvnyu.

Let's take everything at once.

reprocessable:

Axis i was touched by Pythagoras - they brought his theorem in an ancient way.

Rectangular tricycle and trigonometry

For a straight-backed tricycle, check the offensive performance:

The sinus of the gostry kut is used to replace the protruding leg to the hypotenuse

The cosine of a good leg is from the position of an adjacent leg to a hypotenuse.

The tangent of the gosty kut is from the leading leg to the snug leg.

The cotangent of the gooseneck leg is brought from the adjacent leg to the protruding leg.

Once again, all the signs are at the viglyadі:

Tse duzhe zruchno!

Signs of equanimity of upright trikutniks

I. On two legs

II. For the leg and hypotenuse

III. For hypotenuse and gostroma kutu

IV. For a leg and a gostroma kuta

Uwaga! Here it is even more important, the legs of the legs are "vidpovidnі". For example, it will be like this:

Those TRIKUTNIKI NOT RIVNІ, Uninvolved on those who show one by one the same guest kuta.

Needed, schob in both tricytniks, the leg buv will fit, or in both - with.

And, having said, what are the signs of equanimity of upright triketers from the most common signs of equanimity of triketers?

Zazirni in the theme “and I respect the animals for those who, for the equanimity of the“ ordinary ”tricycles, the parity of three elements is required: two sides and a cut between them, two cuts and a side between them or three sides.

And the axis for the equilibrium of rectangular tricycles is to reach two different elements. Great, isn't it?

Approximately the same is the situation with signs of similarity of rectangular tricycles.

Signs of the characteristics of straight-backed tricycles

I. For the Gosting Kuta

II. On two legs

III. For a leg and hypotenuse

Mediana in rectangular tricycle

Why is it so?

It is easy to see the replacement of the rectangular tricycle.

Draw diagonal and clear point - point of overflow of diagonals. What about the rectangular diagonals?

What is the best drink?

Axis i viishlo, scho

- mediana:

Remember the fact! Douzhe dopomagaє!

And it’s more wonderful, so it’s good, it’s true and sound.

What good can be said about the fact that the median was carried out before the hypotenuse, half of the hypotenuse? Let's marvel at the picture

Give me an axis for some reason for "krym that ...".

Wonder at i.

Ale at the junior trikutikiv all kuti rіvnі!

The same can be said about i

And now all at once:

The yak, the cinnamon, can be otrimatized from the tsy "tricky" likeness.

Well for nothing - two formulas for the height of the rectangular tricycle.

Recordable for each side:

For the knowledge of the visoti, the proportion is virishuєmo I write the formula "Visota in a rectangular tricycle":

Well, the axis, now, zasosovuchi and combinuyuchi and znannya zinshih, ty vyrishish be-yak zabadannya with a straight-headed tricycle!

Otzhe, you can zastosuvati butt :.

What will you do now?

I know the proportions and accept the formula to a friend:

The offense of formulas is required for even better memory and storage for that, which is better.

Writable їх more times

Pyfagor's theorem:

The rectangular tricycle has a square of hypotenuzi for the sum of squares of legs :.

Signs of equanimity of straight-backed tricycles:

for two legs:
on the leg and hypotenuse: abo
on the leg and on the side of the gostry kuta: abo
on the leg і of the protylezhny gostry kut: abo
on hypotenuse and gostrom kuti: abo.

Signs of the characteristics of straight-backed tricycles:

one gostry kuta: abo
from the proportions of the two legs:
from the proportions of the leg and the hypotenuse: abo.

Sine, cosine, tangent, cotangent in rectangular tricycle

The sinus of the gostny kut of the rectangular tricycle is called the delivery of the protylny leg to the hypotenuse:
The cosine of a straight leg of a rectangular tricycle is the delivery of an adjacent leg to a hypotenuse:
The tangent of the gostry kut of a rectangular tricycle is called the wear of a protylny leg to a snug one:
The cotangent of a straight leg of a straight leg tricycle is the delivery of an adjacent leg to a protylny leg.

The visota of the rectangular tricycle: abo.

In the rectangular tricycle, the median is drawn from the top of the straight kut, half of the hypotenuse:

The area of the rectangular tricycle:

via catheti: