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Grade 8 L.S. Atanasyan Geometry 7-9 Inscribed and circumscribed circles

О D В С If all sides of the polygon touch the circle, then the circle is said to be inscribed in the polygon. A E A A polygon is said to be inscribed about this circle.

D B C Which of the two quadrilaterals ABC D or AEK D is described? A E K O

D B C A circle cannot be inscribed in a rectangle. A O

D B C What known properties will be useful to us in the study of the inscribed circle? A E O K Tangent property Property of tangent segments F P

D В С In any circumscribed quadrilateral, the sums of opposite sides are equal. A E O a a R N F b b c c d d

D B C The sum of two opposite sides of the circumscribed quadrilateral is 15 cm. Find the perimeter of this quadrilateral. A O No. 695 B C+AD=15 AB+DC=15 P ABCD = 30 cm

D F Find FD A O N ? 4 7 6 5

D B C An isosceles trapezoid is circumscribed about a circle. The bases of the trapezoid are 2 and 8. Find the radius of the inscribed circle. A B C+AD=1 0 AB+DC=1 0 2 8 5 5 2 N F 3 3 4 S L O

D B C The converse is also true. А О If the sums of opposite sides of a convex quadrilateral are equal, then a circle can be inscribed in it. BC + A D = AB + DC

D B C Can a circle be inscribed in the given quadrilateral? A O 5 + 7 \u003d 4 + 8 5 7 4 8

B C A A circle can be inscribed in any triangle. Theorem Prove that a circle can be inscribed in a triangle Given: ABC

K B C A L M O 1) DP: bisectors of the angles of the triangle 2) C OL = CO M, along the hypotenuse and the rest. corner O L \u003d M O Draw from point O perpendicular to the sides of the triangle 3) MOA \u003d KOA, along the hypotenuse and ost. corner MO \u003d KO 4) L O \u003d M O \u003d K O point O is equidistant from the sides of the triangle. Hence, the circle centered at point O passes through the points K, L and M. The sides of triangle ABC are tangent to this circle. So the circle is an inscribed circle ABC.

K B C A A circle can be inscribed in any triangle. L M O Theorem

D B C Prove that the area of ​​the circumscribed polygon is half the product of its perimeter and the radius of the inscribed circle. A No. 69 7 F r a 1 a 2 a 3 r O r ... + K

О D В С If all the vertices of the polygon lie on a circle, then the circle is said to be circumscribed about the polygon. A E A A polygon is said to be inscribed in this circle.

O D B C Which of the polygons shown in the figure is inscribed in a circle? A E L P X E O D B C A E

O A B D C What known properties will be useful to us in the study of the circumscribed circle? Inscribed angle theorem

O A B D In any inscribed quadrilateral, the sum of the opposite angles is 180 0 . C + 360 0

590? 900? 650? 100 0 D A B C O 80 0 115 0 D A B C O 121 0 Find unknown corners of quadrilaterals.

D The converse is also true. If the sum of the opposite angles of a quadrilateral is 180 0 , then a circle can be inscribed around it. 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

B C A A circle can be circumscribed about any triangle. Theorem Prove that it is possible to describe a circle Given: ABC

K B C A L M O 1) DP: perpendicular bisectors to the sides BO \u003d CO 2) B OL \u003d CO L, along the legs 3) COM \u003d A O M, along the legs CO \u003d AO 4) BO \u003d CO \u003d AO, i.e. e. point O is equidistant from the vertices of the triangle. This means that a circle centered at t.O and radius OA will pass through all three vertices of the triangle, i.e. is the circumscribed circle.

K B C A A circle can be circumscribed around any triangle. L M Theorem O

O B C A O B C A No. 702 A triangle ABC is inscribed in a circle so that AB is the diameter of the circle. Find the angles of the triangle if: a) BC = 134 0 134 0 67 0 23 0 b) AC = 70 0 70 0 55 0 35 0

O B C A No. 703 An isosceles triangle ABC with base BC is inscribed in a circle. Find the angles of the triangle if BC = 102 0 . 102 0 51 0 (180 0 - 51 0) : 2 = 129 0: 2 = 128 0 60 / : 2 = 64 0 30 /

O B C A No. 704 (a) A circle with center O is circumscribed about a right triangle. Prove that point O is the midpoint of the hypotenuse. 180 0 diam eter

O B C A No. 704 (b) A circle with center O is circumscribed about a right triangle. Find the sides of the triangle if the diameter of the circle is d and one of the acute angles of the triangle is. d

O C B A No. 705 (a) A circle is circumscribed near a right triangle ABC with right angle C. Find the radius of this circle if AC=8 cm, BC=6 cm. 8 6 10 5 5

O C A B No. 705(b) A circle is circumscribed near a right triangle ABC with a right angle C. Find the radius of this circle if AC=18 cm, 18 30 0 36 18 18

O B C A The sides of the triangle shown in the figure are 3 cm. Find the radius of the circle circumscribed around it. 180 0 3 3

O B C A The radius of the circle circumscribing the triangle shown in the drawing is 2 cm. Find the side AB. 180 0 2 2 45 0 ?

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slide 1

slide 2

Definition: A circle is said to be circumscribed about a triangle if all the vertices of the triangle lie on this circle. If a circle is circumscribed about a triangle, then the triangle is inscribed in the circle.

slide 3

Theorem. A circle can be circumscribed about a triangle, and moreover, only one. Its center is the point of intersection of the midperpendiculars to the sides of the triangle. Proof: Let's draw the perpendicular bisectors p, k, n to the sides AB, BC, AC By the property of the perpendicular bisectors to the sides of the triangle (a wonderful point of the triangle): they intersect at one point - O, for which OA \u003d OB \u003d OS. That is, all the vertices of the triangle are equidistant from the point O, which means that they lie on a circle with center O. This means that the circle is circumscribed near the triangle ABC.

slide 4

Important property: If a circle is circumscribed near a right triangle, then its center is the midpoint of the hypotenuse. R \u003d ½ AB Task: find the radius of a circle circumscribed about a right triangle whose legs are 3 cm and 4 cm.

slide 5

Formulas for the radius of a circle circumscribed about a triangle Task: find the radius of a circle circumscribed about an equilateral triangle whose side is 4 cm. Solution:

slide 6

Problem: An isosceles triangle is inscribed in a circle with a radius of 10 cm. The height drawn to its base is 16 cm. Find the side and area of ​​the triangle. Solution: Since the circle is circumscribed near the isosceles triangle ABC, the center of the circle lies at the height BH. AO = VO = CO = 10 cm, OH = VN - VO = = 16 - 10 = 6 (cm) AC = 2AH = 2 8 = 16 (cm), SABC = ½ AC WH = ½ 16 16 128 (cm2)

Slide 7

Definition: A circle is said to be circumscribed about a quadrilateral if all the vertices of the quadrilateral lie on the circle. Theorem. If a circle is circumscribed near a quadrilateral, then the sum of its opposite angles is equal to 1800. Proof: Another formulation of the theorem: in a quadrangle inscribed in a circle, the sum of opposite angles is equal to 1800.

Slide 8

Converse theorem: if the sum of the opposite angles of a quadrilateral is 1800, then a circle can be circumscribed around it. Proof: № 729 (textbook) Around which quadrilateral is it impossible to circumscribe a circle?

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In which figure is a circle inscribed in a triangle?

If a circle is inscribed in a triangle,

then the triangle is circumscribed about the circle.

Theorem. A circle can be inscribed in a triangle, and moreover, only one. Its center is the intersection point of the bisectors of the triangle.

Given: ABC

Prove: there exists Osp.(O; r),

inscribed in a triangle

Proof:

Let's draw the bisectors of the triangle: AA 1, BB 1, SS 1.

By property (remarkable point of the triangle)

bisectors intersect at one point - O,

and this point is equidistant from all sides of the triangle, i.e.:

OK \u003d OE \u003d OR, where OK AB, OE BC, OR AC, then

O is the center of the circle, and AB, BC, AC are tangents to it.

So the circle is inscribed in ABC.

Given: Okr. (O; r) is inscribed in ABC,

p \u003d ½ (AB + BC + AC) - half-perimeter.

Prove: S ABC = p r

Proof:

connect the center of the circle with the vertices

triangle and draw the radii

circles at points of contact.

These radii are

heights of triangles AOB, BOC, COA.

S ABC = S AOB + S BOC + S AOC = ½ AB r + ½ BC r + ½ AC r =

= ½ (AB + BC + AC) r = ½ p r.

Task: into an equilateral triangle with a side of 4 cm

inscribed circle. Find its radius.

Derivation of the formula for the radius of a circle inscribed in a triangle

S = p r = ½ P r = ½ (a + b + c) r

2S = (a + b + c) r

The desired formula for the radius of a circle,

inscribed in a right triangle

- legs, c - hypotenuse

Definition: A circle is said to be inscribed in a quadrilateral if all sides of the quadrilateral touch it.

In which figure is a circle inscribed in a quadrilateral?

Theorem: if a circle is inscribed in a quadrilateral,

then the sums of opposite sides

quadrilaterals are equal ( in any described

quadrilateral sum of opposites

sides are equal).

AB + SK = BC + AK.

Inverse theorem: if the sums of opposite sides

convex quadrilateral are equal,

then a circle can be inscribed in it.

Task: in a rhombus, the acute angle of which is 60 0, a circle is inscribed,

whose radius is 2 cm. Find the perimeter of the rhombus.

Solve problems

Given: Okr. (O; r) is inscribed in ABSK,

P ABSC = 10

Find: BC + AK

Given: ABSM is described around approx. (O; r)

BC=6, AM=15,

OA=OB O b => OB=OC => O perpendicular bisector to AC => approx. ABC can describe a circle ba =>OA=OC =>" title="(!LANG:Theorem 1 Proof: 1) a is the perpendicular bisector to AB 2) b is the perpendicular bisector to BC 3) ab=O 4) O a = > OA=OB O b => OB=OC => O perpendicular bisector to AC => approx. ABC can describe a circle ba =>OA=OC =>" class="link_thumb"> 8 !} Theorem 1 Proof: 1) a is the perpendicular bisector to AB 2) b is the perpendicular bisector to BC 3) ab=O 4) O a => OA=OB O b => OB=OC => O the perpendicular bisector to AC => about tr. ABC can describe a circle ba =>OA=OC => OA=OB O b => OB=OC => O perpendicular bisector to AC => approx. ABC can describe a circle ba =>OA=OC =>"> OA=OB O b => OB=OC => O perpendicular bisector to AC => near tr. ABC can describe a circle ba =>OA=OC =>"> OA=OB O b => OB=OC => O perpendicular bisector to AC => approx. ABC can describe a circle ba =>OA=OC =>" title="(!LANG:Theorem 1 Proof: 1) a is the perpendicular bisector to AB 2) b is the perpendicular bisector to BC 3) ab=O 4) O a = > OA=OB O b => OB=OC => O perpendicular bisector to AC => approx. ABC can describe a circle ba =>OA=OC =>"> title="Theorem 1 Proof: 1) a is the perpendicular bisector to AB 2) b is the perpendicular bisector to BC 3) ab=O 4) O a => OA=OB O b => OB=OC => O the perpendicular bisector to AC => about tr. ABC can describe a circle ba =>OA=OC =>"> !}

Properties of a triangle and a trapezoid inscribed in a circle obtuse-angled tr-ka, does not lie in the tr-ke