The circumference is described as the tricuputide. Described col 1 inscribed and described col Presentation

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"Tables with geometry" - Tables. Multiplying the vector by the Axis number is central symmetry. One hundred stake Central and inscribed kuti Inscribed and described in the circle Concept of vector Folding and vydnіmanya of vectors. Location: Orthocutanea Parallelogram and trapezoid Rectum, rhombus, square Area of ​​Orthocutaneum Area of ​​tricutaneous, parallelogram and trapezium Pythagorean theorem Similar tricutnuts Signs of the similarity of tricutnuts Interrelationship between the sides and cutata of the recticutaneous three Kutnik Vzaem.

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8th grade L.S. Atanasyan Geometry 7-9 Inscribed and described stake

O D B C Since all sides of the rich-kutnik stick together, then the circle is called inscribed in the rich-kutnik. A E And the rich cutthroat is called a description of the white stake.

D B C Which of the two four cuticles ABC D or AEK D є will we describe? A E K O

D B C It is not possible to enter a colo into the rectangular plant. A O

D Q What kind of power do we need when installing a registered stake? A E O Power of the military Power of the branches of the military F P

D Any described chotirikutnik has sums of the protilegny sides of the river. A E About a a R N F b b c c d d

D C The sum of the two sides of the described chotirikutnik is 15 cm. Find the perimeter of this chotirikutnik. A Pro No. 695 C+AD=15 AB+DC=15 P ABCD = 30 cm

D F Know FD A N ? 4 7 6 5

D Y C The right-sided trapezium is described white. Substitute the trapezoid to match 2 and 8. Find the radius of the inscribed stake. A C+AD=1 0 AB+DC=1 0 2 8 5 5 2 N F 3 3 4 S L O

D B C It is true and the turning point is firm. A O If the protilegal sides of the convex chotirikutnik become more mature, then you can enter a colo into the new one. ND + A D = AB + DC

D U S Chi can you enter Chotirikutnik in Kolo? A Pro 5 + 7 = 4 + 8 5 7 4 8

In C A In any trikutnik you can enter a colo. Theorem Show that a colo can be entered into a trikutnik Given: ABC

K B S A L M O 1) DP: bisectors of the tricutaneous cuticles 2) C OL = CO M, s hypotenuse and zoom. kuti O L = M About Draw from the point About the perpendiculars to the sides of the tricutaneous 3) MOA = KOA, from the hypotenuse and zoom. kuti MO = KO 4) L O = M O = K O point O is evenly removed from the sides of the tricubitule. Then, around the center, it passes through points K, L and M. The sides of the trikutnik ABC are stuck together with this stake. Well, ABC is written in there.

K S A In any trikutnik you can enter a colo. L M About the Theorem

D Y C Assume that the area of ​​the described rich tuft is equal to half of its perimeter to the radius of the inscribed stake. A No. 69 7 F r a 1 a 2 a 3 r O r ... + K

O D BC If all the vertices of the rich-cutlet lie on a stake, then the stake is called a description of the rich-cutlet. A E And the rich cutlet is called inscribed in tse colo.

O D B H Which of the rich cuttlefish depicted in the baby is included in the circle? A E L P X E O D B C A E

About A B D C What kind of power do we need to describe the stake? Theorem about Vugill's inscriptions

In any inscribed chotirikutnik, the sum of the protilegal kuti is equal to 180 0. W + 360 0

59 0 ? 90 0 ? 65 0 ? 100 0 D A B C O 80 0 115 0 D A B C O 121 0 Find out the unknown corners of the cuties.

D That is the correct turning point. If the sum of the protilegal kuti of the chotiricutnik is more than 180 0, then you can also enter a colo. A B C O 80 0 100 0 113 0 67 0 O D A B C 79 0 99 0 123 0 77 0

In S A Bil, any trikutnik can be described as a colo. Theorem Bring what can be described given: ABC

K B C A L M O 1) DP: perpendicular bisectors to the sides VO = CO 2) B OL = COL, behind the legs 3) COM = A O M, behind the legs CO = AO 4) VO = CO = AT, i.e. e. point O is equally distant from the tops of the tricutaneous ridge. Ozhe, near the center incl. and with radius OA will pass through the three vertices of the tricutaneous, then. Let's describe the stake.

K U S A Bill of any kind can be described as a colo. L M Pro Theorem

О ВС А О ВС А No. 702 The tricutule ABC is inscribed on the stake so that AB is the diameter of the stake. Find the tricutaneous cuticle, which is: a) BC = 134 0 134 0 67 0 23 0 b) AC = 70 0 70 0 55 0 35 0

OVS No. 703 At the number of inscriptions of the tricutaneous ABC with the base of the BC. Find the tricutaneous kuti, as BC = 1020. 102 0 51 0 (180 0 – 51 0) : 2 = 129 0: 2 = 128 0 60 / : 2 = 64 0 30 /

OBC No. 704 (a) The circle with center O describes the bile of the rectum tricutaneum. Let us know that point O is the middle of the hypotenuse. 180 0 d i a meter

OBC No. 704 (b) The circle with center O describes the bile of the rectum tricutaneum. Find the sides of the trikutnik, where the diameter of the stake is similar to d, and one of the sharp cutlets of the tricubitus is similar. d

OSAV No. 705 (a) Bіlya of the straight cutter ABC with straight cut C is described colo. Find the radius of this stake if AC = 8 cm, BC = 6 cm. 8 6 10 5 5

OSAV No. 705(b) Bіlya of the straight cut tricutaneous ABC with straight cut C is described colo. Find the radius of this stake, if AC = 18 cm, 18 30 0 36 18 18

O B S A The side sides of the tricube depicted in the baby are 3 cm long. Find the radius of the described white stake. 180 0 3 3

O B C A The radius of the stake described by the tricuputin, depicted on the chair, is more than 2 cm. Find the side AB. 180 0 2 2 45 0 ?

Behind the topic: methodical developments, presentations and notes

The pre-lesson presentation includes the identification of basic understandings of how to create a problem situation, as well as the development of students' creative abilities.

Work program for an elective course in geometry “Solving planimetric tasks on an inscribed and described stake” 9th grade

Statistical data from the analysis of the results of the study show that the smallest number of correct answers are provided by geometric studies. The planimetry command must be turned on before...

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Captions before slides:

Described colo

Meaning: the circle is called a description of the tricuputnum, since all the vertices of the tricuputin lie on this stake. For which little colo is described the white of the trikutnik: 1) 2) 3) 4) 5) When the white of the colo is described for the trikutnik, then the trikutnik is written in the colo.

Theorem. The bile duct can be described in more than one way, and even more so. This center is the point of the crossbar of the middle perpendiculars to the sides of the tricut. A B C Given: АВС Bring: існє Ур.(О; r), the word АВС is described. Proof: Let's draw the median perpendiculars p, k, n to the sides AB, BC, AC. Under the power of the median perpendiculars to the sides of the tricutellum (the miracle point of the tricucutineum): the smells move at one point - O for some OA = OB = OS. That is, all the vertices of the tricupull are located at the same distance from point O, so they lie on a ring with center O. This means that the circle is described for the tricuput ABC. About n p k

Power is important: As described earlier, the tricutaneous rectum, its center is the middle of the hypotenusus. O R R C A B R = ½ AB Task: find out the radius of the stake described by the rectum tricucutineum, the legs of which are 3 cm and 4 cm. The center of the stake described by the obtuse cuticle, lie in the tricucutineum position.

a b c R R = Formulas for the radius of the described bicuspid stake Problem: find the radius of the equilateral tricucular stake, the side of which is 4 cm. Solution: R = R = , Version: cm (cm)

Design: in the colo, the radius of which is 10 cm, inscriptions of the equifemoral tricuputin. The height, drawn to the base, is 16 cm. Find out the side and area of ​​the tricupus. A B S O N Rishennya: T. to. the circle is described around the abdomen ABC, the center of the circle lies at the height of BH. AO = VO = CO = 10 cm, VIN = VN - VO = = 16 - 10 = 6 (cm) AON - straight-cut, AO 2 = AN 2 + AN 2, AN 2 = 10 2 - 6 2 = 64, AN = 8 cm AVN - straight-cut, AB 2 = AN 2 + VN 2 = 8 2 + 16 2 = 64 + 256 = 320, AB = (cm) AC = 2AN = 2 8 = 16 (cm), S ABC = ½ AC · VN = ½ · 16 · 16 = 128 (cm 2) Type: AB = cm S = 128 cm 2 Know: AB, S ABC Given: ABC-r / b, VN AC, VN = 16 cm Env. ; 10 cm) described in ABC

Meaning: the stake is called a description of the bile of the chotiricutnik, since all the tops of the chotiricutnik lie on the stake. Theorem. Since a colo is described close to the cuticle, then the sum of its protilegous cutives is equal to 180 0 . Proof: The circle is described by ABC D, then A, B, C, D are inscribed, which means A + C = ½ BCD + ½ BAD = ½ (BCD + BAD) = ½ · 360 0 = 180 0 B+ D = ? C = B + D = 180 0 Another formulation of the theorem: for the inscribed in the bell of the bell, the sum of proliferative kuti is equal to 180 0 . A B C D About

The reversal theorem: if the sum of the protimal cutiva of the chotiricutnik is equal to 180 0, then the circle can be described. Given: ABC D, A + C = 180 0 A B C D O Bring: Surroundings (O; R) described by ABC D Proof: No. 729 (podruchnik) It is impossible to describe the colo of what kind of chotirikutnik?

Nasledok 1: the white of any rectilinear can be described as a circle, its center is the point of the crossbar of the diagonals. Assignment 2: with an equal femoral trapezius, you can describe a colo. A B C K

Unraveling tasks 80 0 120 0 ? ? A B C M K N O R E 70 0 Know the kuti of the chotirikutnik RKEN: 80 0

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