The formula for the radius of the inscribed stake through the sides. Radius of the inscribed stake, formulas, tasks. Colo, inscribed in a rectangular tricycle

It’s like it’s inscribed in the middle of the kut, and it’s inscribed in the whole kut. The center of such an inscribed stake should be stitched on bisectrix kuta.

As soon as you lie down in the middle of the opuky bagatokutnik and stick around the sides, you will be called inscribed in the opuky bagatokutnik.

Colo, inscribed in the trikutnik

Kolo, inscribed in the trikutnik, stick on the skin side of the lishe figurine in one point. In one trikutnik you can write one or more leashes.

The radius of such a circumference will be found in the offensive parameters of the tricycle:

1. Dovzhin storin trikutnik.
2. Yogo area.
3. Yogo perimeter.
4. The size of the tricytes' kutiv.

In order to count the radius of the inscribed stake in the trikutnik, do not depend on the nobility of all the parameters, some of the stench combined with oneself through trigonometric functions.

Calculation for additional help for a perimeter

1. Yakshcho see all sides geometric figurines(Significantly їх letters a, b and c), then calculate the radius to arrive at a square root path.
2. Proceeding to the calculation, it is necessary to add one more change to the due date - for example, the perimeter (p). Yogo can be recovered, having sklavi all dozhini and trimmed the sum by 2. p = (a + b + c) / 2. With this rank, you can literally simplify the formula for knowing the radius.
3. As a whole, the formula is guilty of including the sign of the radical, for which is placed dib, the denominator of the fraction will be the value of the perimeter p.
4. The numeral given to the fraction will be a solid value (p-a) * (p-b) * (p-c)
5. In such a rank, a new form of the formula for representations in such a rank: r = √ (p-a) * (p-b) * (p-c) / p).

The calculation of the area of ​​the tricycle

I’ll see us tricycle area and do the genie of all the sides, let us know the radius to circle us around, not letting go of the roots.

1. For the cob, it is necessary to increase the size of the area.
2. The result lasts for the sum of all parties. Todi the formula will be in the offensive rank: r = 2 * S / (a ​​+ b + c).
3. As fast as the value is on the perimeter, you can make a simple formula: r = S / p.

Role for additional trigonometric functions

Just as in the minds of the presence of the same number of sides, the size of the prototype codend and the perimeter, it is possible to speed up trigonometric function- tangent. The formula for rozrahunku bude mati nasty viglyad is vivid:

r = (P / 2- a) * tg (α / 2), de r is the radius, P is the perimeter, and is the value of one side, α is the value of the opposite side, and kuta.

Radius cola, as it is necessary to be entered into the correct tricycle, can be found behind the formula r = a * √3 / 6.

Colo, inscribed in a rectangular tricycle

V rectangular tricycle you can write only one colo... The center of such a circle will instantly serve as a point of overflow of all bisectors. Tsya geometric figure of ma deyakі vіdmіnnі rice, as it is necessary to vrahuvati, calculating the radius of the inscribed stake.

1. For the cob, it is necessary to vibrate the rectangular tricycle with the given parameters. You can stay taku figuru behind the size of one side and the size of two kutiv or on two sides and kutu between sides. All parameters are guilty of being mentioned in the mind of the manager. Tricutnik is known as yak ABC, and C is the top straight kuta... Catheti at all are known as wintry, aі b, And hypothenuse - change s.
2. To induce the classic formula and the calculation of the stake radius, it is necessary to consider all the sides described in the mind of the figurine and to calculate the perimeter using them. As far as the minds are given the size of two legs, they can be used to calculate the value of the hypothenuse, which is based on the theorems of Pythagoras.
3. As in the mind, the size of one leg and one cut is given, it is necessary to be visibly attached to the cut or opposed. The first hypothesis of hypothesis is behind the additional theorem of sinuses: s = a / sinCAB, The other has a problem with the cosine theorem s = a / cosCBA.
4. If all the sizes of the viconians and the magnitudes of all sides are visible, then the perimeter for the formula described by the visce is known.
5. Knowing the value of the perimeter, you can know the radius. The formula is drib. The numerator is the value for the perimeter and the skin of the sides, and the denominator is the value for the perimeter.

Slid respect, that the number of the given formula is an indicator of the area. In general, the formula for the knowledge of the radius is to say goodbye - enough space for a perimeter.

Visually, the area of ​​a geometrical figure is possible in the same way as if the leg was hurt. Behind the sum of squares of the number of legs there is a hypotenuse, it is possible to calculate it by a perimeter. You can calculate the area by multiplying one by one the size of the legs and separated by 2.

Just as in the minds it is given to the genie and cathetes and hypotenuse, but the number of radii is possible by even simpler formulas: for the whole number of cathetes are stored, from the specified number of hypotenuse. The result is necessary to distribute navpil.

Video

You can see the third video that you know the radius of the stake inscribed in the tricycle.

Colo, inscribed in the trikutnik

Іsnuvannya stake, inscribed in a trikutnik

Nagadaєmo name bisektrix kuta .

value 1 .bisector kuta I name promin, what kind of dilev kut on two separate parts.

theorem 1 (The main power of the bisector kuta) ... The skin point of the bisector kuta is located on one and the same side from the side of the kuta (Fig. 1).

Small. 1

Dovedennya D , To lie on the bisektrix kutaBAC , і DE і DF on the side of the kuta (Fig. 1).rectangular tricycles ADF і ADE рівні , Oskilki they have equal gostri kutiDAF і DAE , A hypotenuse AD - zagalna. already,

DF = DE,

it’s necessary to bring it.

Theorem 2 (vortex theorem before Theorem 1) ... If you are deyaka, then lie on the bisector kuta (Fig. 2).

Small. 2

Dovedennya ... I can see the pointD To lie in the middle of the kutaBAC and be located on one and on the other side of the side. Opportunity from pointD perpendiculars DE і DF on the side of the kuta (Fig. 2).rectangular tricycles ADF і ADE рівні , Oskіlki they have equal cathetiDF і DE , A hypotenuse AD - zagalna. already,

it’s necessary to bring it.

value 2 ... circle name circumference, inscribed in the kut , Yaksho won a side cut.

Theorem 3 ... As the circle is inscribed in the kut, then it is seen from the top of the kut to the points of the dotted line from the sides of the kut rivn.

Dovedennya ... hello point D - the center of the stake inscribed in the kutBAC , A points E і F - dots dots kola from the sides of the kuta (Fig. 3).

fig. 3

a , b , c - sides of the trikutnik,
 S -area,

r – radius of the inscribed stake,
 p - napivperimeter

.

Marvel at the formulas

a – bichna side tricycle tricycle , 
 b - the foundation, r – radius of the inscribed stake

a r – radius of the inscribed stake

Marvel at the formulas

,

de

,

then, in times of a femoral tricycle, if

otrimmo

well i bulo is required.

Theorem 7 ... For fairness

de a - side of an equilateral tricycle,r – the radius of the inscribed stake (Fig. 8).

Small. eight

Dovedennya .

,

then, in the case of an unilateral trike, if

b = a,

otrimmo

well i bulo is required.

respect ... I recommend that you put in the right formula for the radius of the stake, inscribed in the equilateral tricycle, without the middle, tobto without the vikoristanny formulas for radiius k_l, inscribed in a previlny trikutnik or in a trikutnik.

Theorem 8 ... For a rectangular tricycle, it is fair to say

de a , b - legs of a rectangular tricycle, c – hypotenuse , r – radius of the inscribed stake.

Dovedennya ... Little ones are visible 9.

Small. nine

oskilki chotirikutnikCDOF є , At which sideDO і OF rіvnі, then tsey rectangular - . already,

CB = CF = r,

By virtue of Theorems 3, it is true that

Otzhe, accept the same until uvagi, take it away

well i bulo is required.

Dobirka zavdan on the theme "Kolo, inscribed in a trikutnik".

1.

Colo, inscribed in the tricycle, divide in the dotted one from the other sides into two branches, even if the sides are 5 and 3, rakhuyuchi from the top, the opposite base. Know the perimeter of the tricycle.

2.

3

For a tricycle ABC AC = 4, BC = 3, Kut C door 90º. Know the radius of the inscribed stake.

4.

The legs of the rectangular tricyx tricycle, rivni 2+. Know the radius of the stake inscribed in the tricycle.

5.

Radius of a stake, inscribed in a raspberry tricycle, door 2. Know the hypotenuse from a tricycle tricycle. In the example, it should be shown with (-1).

Probably a number of buildings with ЄDI with solutions.

Radius of a stake, inscribed in a straight-line tricycle, dorivnyu. Know the hypothesis from the tricycle. In the case of the message.

Trikutnik rectangular і wedges. This means that both sides are the same. Come on leather cathetus for the door... Todi hypotenuse for home.

There are two ways to write the area of ​​the ABC tricycle:

Adopting tsi virazi, otrimaєmo, scho... scraps, Otrimuєmo, scho... Todi.

We will write down.

as follows:.

Zavdannya 2.

1. For more than two double sides 10cm and 6cm (AB and BC). Know the radius described and inscribed
Zavdannya virіshuєtsya independently from comments.

Decision:

V.

1) Know:
2) Bring:i know SK
3) Know: the radius is described and inscribed

Decision:

Zavdannya 6.

R adius of a circle inscribed in a square door... Know the radius of the stake described around a square.given :

know: OS =?
Decision: In this particular type of vypadannya can be violated by speeding either the Pythagorean theorem, or the formula for R.

Zavdannya 7.

Radius of the stake, inscribed in the ravnobrenyi upright trikutnik, dorivnyu 2. Know the hypotenuses tsyogo trikutnik. At the top of the list.

S - tricycle area

We do not know any side of the tricycle, no area. Apparently the catheti yak x, that is, the hypothesis will be available:

And the area of ​​the tricycle will be suitable for 0.5 x 2 .

to mean

In such a rank, the hypothesis will be dorivnyu:

At the top it is necessary to write down:

View: 4

Zavdannya 8.

For tricycle ABC AC = 4, BC = 3, cut C door 90 0. Know the radius of the inscribed stake.

Skoristaєmosya with the formula radius cola inscribed in the trikutnik:

de a, b, c - sides and tricut

S - tricycle area

Two sides are visible (tse catheti), we can count the third (hypotenuzi), we can also count the area.

According to Pythagoras' theorem:

I know the area:

In this rank:

View: 1

Zavdannya 9.

Bichni sides of the femoral tricycle equal to 5, the basis of the road 6. Know the radius of the inscribed stake.

Skoristaєmosya with the formula radius cola inscribed in the trikutnik:

de a, b, c - sides and tricut

S - tricycle area

See all sides, numbered and area. We can know for Heron's formula:

Todi

A rhombus is a parallelogram, at which all sides are equal. Otzhe, vіn usadkovu all the powers of the parallelogram. And itself:

• The diagonals of the rhombus are mutually perpendicular.
• Diagonals of a rhombus є bisectors of the innermost coils.

The surroundings can be entered into the chotirikutnik todі and only todі, if the sum of the other sides is equal.
Otzhe, in any rhombus you can write a colo. The center of the inscribed stake is located at the center of the overflow of the diagonals of the rhombus.
The radius of the inscribed stake in the rhombus can be seen in several ways

1 way Radius of an inscribed stake in a rhombus through the top

The height of the rhombus is equal to the diameter of the inscribed stake. The price is due to the power of the rectangle, which sets the diameter of the inscribed stake and the height of the rhombus - at the rectangle of the opposite sides of the rivn.

The same formula for the radius of an inscribed stake in a rhombus through the height:

2 way Radius of an inscribed stake in a rhombus through diagonals

The area of ​​the rhombus can be drawn through the radius of the inscribed stake
, de R is the perimeter of the rhombus. You know, the perimeter of the tse sum of all sides of the chotirikutnik maєmo P = 4× a. Todi
Ale the area of ​​the rhombus, so also the half-dozen diagonals
Adopt the right part of the area formulas,
As a result, we can recognize a formula that allows us to calculate the radius of the inscribed stake in a rhombus through diagonals

The butt of a rozrakhunku radius cola inscribed in a rhombus, when viewed from a diagonal
Know the radius of a stake inscribed in a rhombus, when it is vidomo, when it comes to diagonals 30 cm and 40 cm
hey ABCD-rombus, todі ACі BD yogo diagonal. AC = 30 cm , BD= 40 cm
hello point About- tse center inscribed in a rhombus ABCD circumference, so you will also be a point of overturning your diagonals, which will add to your navpil.


because the diagonal of the rhombus is overturned with a straight cut, then the tricycle AOB rectangular. Todi by Pyfagor's theorem
, Pidstavlyaєmo in the formula before the rejection of the value

AB= 25 cm
Having stuck the formula for the radius of the described stake in a rhombus earlier, I will

3 way The radius of the inscribed stake in a rhombus through the lines m and n

Speck F- dot dotted with cola on the side of the rhombus AFі Bf... hey AF =m, BF = n.
Speck O- the center of the overflow of the diagonals of the rhombus and the center of the inscribed in the new stake.
trikotnik AOB- straight-cut, so like a diagonal rhombus is overturned by a straight cut.
To that, with a radius, we will carry out a dotted stake to the point. already OF- the height of the tricot AOB to hypotenuse. Todi AFі BF - projection of cathetes to hypotenuse.
The head is in the rectangular tricycle, lowered to the hypotenuse є average proportional and the projections of the legs to the hypotenuse.

The formula for the radius of the inscribed stake in a rhombus through the cuts to the square root from the add-on of the cich to the coke, on the side of the rhombus the dot of the dot to the coke

Clearly colo, inscribed in the trikutnik (fig. 302). Nagadaєmo, scho її center About to sit on the cross-bisectors of the inner corners of the tricycle. Відрізки ОА, ОВ, ОС, scho z'єdnuyut About the tops of the tri-runner ABC, rozib'yut the trikutnik into three triplets:

AOV, VOS, SOA. The sagging of the skin from the tricycle of the tricycle in the direction of the radius

The area of ​​the entire trikotnik S for the road is the sum of the three areas:

de - napіvperimeter of a tricycle. Zvidsi

The radius of the inscribed stake is taken from the area of ​​the tricycle to the first perimeter.

To reject the formula for the radius of the described stake of the trikutnik, we will bring the proposition.

Theorem a: Any tricycle has a side to the diameter of the described stake, multiplied by the sine of the prototype kut.

Delivered. Clearly satisfied tricycle ABC I will be described near the nye, the radius of which is signifi- cant through R (Fig. 303). Nekhai A - gostry kut tricutnik. Carrying out the radius of the OB, the OS of the circle and dropping from the center About the perpendicular OK to the side of the VS of the tricycle. Splendidly, how the cut and the tricycle are shaped by half the arc of the BC, for which the cut of the VOS є the central cut. Zvidsey is visible, scho. The fact of the rectangular trikulet of the SOC is known, or else, it was necessary to bring it up.

Hovering fig. 303 and the world gostry kuta trikutnik; It’s not important to prove it for the types of direct and blunt kuta (the reader tries to play independently), but it is possible to prove the sinus theorem (218.3). So yak maє buti stars

The sinus theorem can also be written in. viglyadі

i will write down in form (218.3) yes for

The radius of the described stake is used to create three sides of the tricycle to its quadrupled area.

Zavdannya. Know the sides of the femoral tricycle, where it is inscribed and described by the circumference of the

Decision. Let us write the formulas for turning the radii of the inscribed and described tricycle:

For a femoral tricycle from the toe side and before the area, turn the formula

anyway, speeding up the multiplier to the difference between zero, matimemo

to build up to a square ryvnyannya

Wono has two solutions:

Submitting a substitute for a viraz to be-like a rivnyan for any R, we know that there are two remaining answers to our default:

right

1. The head of the rectus tricot, carried out from the top of the straight cut, delnt hypotenuse to know the performance of the skin catheti to hypotenuse.

2. Submit an equal trapezium, described close to the circumference, equal a and b. Know the radius cola.

3. Two stakes are worthy of the name. Outside of the dots to the point of the center line up to 30 °. Dovzhina is dotted dots dots dots dots 108 cm. Know radiusi kil.

4. The legs of the rectangular tricycle runner a і b. Know the area of ​​the tricycle, the sides of which serve as the height and median of the given tricycle, drawing from the top of the straight cut, and from the top of the hypotenuse between the points and the overflow to the hypotenuse.

5. The sides of the tricycle are made up 13, 14, 15. To know the projection of the skin from them on the two sides.

6. At the tricycle, see the side and the side. Know the side b and with.

7. See the two sides of the tricycle and the median Know the third side of the tricycle.

8. Given two sides of the tricycle and the cut and between them: Know the radii of the inscribed and described cil.

9. On the side of the tricycle a, b, c. Why are the odds, on how the stench break the dots of the dots of the inscribed stake on the sides of the tricycle?