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 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="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
To use the preview of presentations, create a Google account (account) and sign in: https://accounts.google.com
Slides captions:
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 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 sharp corners triangle is equal. 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 ?
On the topic: methodological developments, presentations and notes
The presentation for the lesson includes definitions of basic concepts, the creation of a problem situation, as well as the development creativity students....
The work program for the elective course in geometry "Solution of planimetric problems on inscribed and circumscribed circles" Grade 9
Results analysis statistics conducting the exam they say that the smallest percentage of correct answers is traditionally given by students to geometric problems. Planimetry tasks included in...
To use the preview of presentations, create a Google account (account) and sign in: https://accounts.google.com
Slides captions:
Circumscribed circle
Definition: A circle is said to be circumscribed about a triangle if all the vertices of the triangle lie on this circle. In which figure is the circle circumscribed near the triangle: 1) 2) 3) 4) 5) If the circle is circumscribed near the triangle, then the triangle is inscribed in the circle.
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. A B C Given: ABC Prove that there exists an Osp.(O; r) described around ABC. 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. O n p k
Important property: If a circle is circumscribed near a right triangle, then its center is the midpoint of the hypotenuse. O R R C A B R = ½ AB Task: find the radius of a circle circumscribed about a right triangle whose legs are 3 cm and 4 cm. The center of the circle circumscribed about an obtuse triangle lies outside the triangle.
a b c R R = 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: R = R = , Answer: cm (cm)
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. A B C O N 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) AON - rectangular, AO 2 = AH 2 + AH 2, AH 2 = 10 2 - 6 2 = 64, AH = 8 cm ABN - rectangular, AB 2 \u003d AN 2 + VN 2 \u003d 8 2 + 16 2 \u003d 64 + 256 \u003d 320, AB \u003d (cm) AC \u003d 2AN \u003d 2 8 \u003d 16 (cm), S ABC \u003d ½ AC VN \u003d ½ 16 16 \u003d 128 (cm 2) Answer: AB \u003d cm S \u003d 128 cm 2, Find: AB, S ABC Given: ABC-r / b, VN AC, VN \u003d 16 cm Okr. (O ; 10 cm) described near ABC
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 180 0 . Proof: Since the circle is circumscribed about ABC D, then A, B, C, D are inscribed, which means A + C \u003d ½ BCD + ½ BAD \u003d ½ (BCD + BAD) \u003d ½ 360 0 \u003d 180 0 B+ D = ½ ADC + ½ ABC = ½ (ADC+ ABC) = ½ 360 0 = 180 0 A + C = B + D = 180 0 C = B + D = 180 0 Another formulation of the theorem: in a quadrilateral inscribed in a circle, the sum of opposite angles is 180 0 . A B C D O
Inverse theorem: if the sum of the opposite angles of a quadrilateral is 180 0, then a circle can be circumscribed around it. Given: ABC D, A + C = 180 0 A B C D O
Corollary 1: a circle can be circumscribed near any rectangle, its center is the intersection point of the diagonals. Corollary 2: A circle can be circumscribed about an isosceles trapezoid. A B C K
Solve problems 80 0 120 0 ? ? A B C M K N O R E 70 0 Find the angles of the quadrilateral RKEN: 80 0
