Presentation on geometry "inscribed and circumscribed circle". A circle circumscribed about a triangle is the sum of the opposite sides
In which picture is a circle inscribed in a triangle?
If a circle is inscribed in a triangle,
then the triangle is circumscribed about a circle.
Theorem. You can inscribe a circle in a triangle, and only one. Its center is the point of intersection of the bisectors of the triangle.
Given by: ABC
Prove: there is Env.(O; r),
inscribed in a triangle
Proof:
Let's draw the bisectors of the triangle: AA 1, BB 1, СС 1.
By property (remarkable point of the triangle)
bisectors intersect at one point - Oh,
and this point is equidistant from all sides of the triangle, i.e.:
OK = OE = OR, where OK AB, OE BC, OR AC, which means
O is the center of the circle, and AB, BC, AC are tangents to it.
This means that the circle is inscribed in ABC.
Given: Environment (O; r) is inscribed in ABC,
p = ½ (AB + BC + AC) – semi-perimeter.
Prove: S ABC = p r
Proof:
connect the center of the circle with the vertices
triangle and draw the radii
circles at the points of contact.
These radii are
altitudes 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: in an equilateral triangle with a side of 4 cm
circle is inscribed. 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 required formula for the radius of a circle is
inscribed in a right triangle
- legs, c - hypotenuse
Definition: A circle is called 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.
Converse theorem: if the sums of opposite sides
convex quadrilateral are equal,
then you can fit a circle into it.
Problem: a circle is inscribed in a rhombus whose acute angle is 60 0,
whose radius is 2 cm. Find the perimeter of the rhombus.
Solve problems
Given: Env.(O; r) is inscribed in ABCC,
R ABCC = 10
Find: BC + AK
Given: ABCM is described about Environ.(O; r)
BC = 6, AM = 15,
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Slide captions:
Circumcircle
Definition: a circle is called circumscribed about a triangle if all the vertices of the triangle lie on this circle. In which figure is a circle described around a triangle: 1) 2) 3) 4) 5) If a circle is described around a triangle, then the triangle is inscribed in the circle.
Theorem. Around a triangle you can describe a circle, and only one. Its center is the point of intersection of the perpendicular bisectors to the sides of the triangle. A B C Given: ABC Prove: there is an Environment (O; r) described near ABC. Proof: Let us draw perpendicular bisectors p, k, n to the sides AB, BC, AC. According to the property of perpendicular bisectors to the sides of a triangle (a remarkable point of a triangle): they intersect at one point - O, for which OA = OB = OC. That is, all the vertices of the triangle are equidistant from point O, which means they lie on a circle with center O. This means that the circle is circumscribed about triangle ABC. O n p k
Important property: If a circle is circumscribed about a right triangle, then its center is the midpoint of the hypotenuse. O R R C A B R = ½ AB Problem: find the radius of a circle circumscribed about a right triangle, the legs of which are 3 cm and 4 cm. The center of a circle circumscribed about an obtuse triangle lies outside the triangle.
a b c R R = Formulas for the radius of a circle circumscribed by a triangle Task: find the radius of a circle circumscribed by 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 lateral side and area of the triangle. A B C O N Solution: Since the circle is circumscribed about 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 = AN 2 + AN 2, AN 2 = 10 2 – 6 2 = 64, AN = 8 cm ABN - rectangular, 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 · ВН = ½ · 16 · 16 = 128 (cm 2) Answer: AB = cm S = 128 cm 2, Find: AB, S ABC Given: ABC-r/b, VN AC, VN = 16 cm Surround (O ; 10 cm) is 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 around a quadrilateral, then the sum of its opposite angles is equal to 180 0. Proof: Since the circle is circumscribed about 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 = ½ ADC + ½ ABC = ½ (ADC+ ABC) = ½ 360 0 = 180 0 A + C = B + D = 180 0 Given: Environment (O; R) is described around ABC D Prove: So A + 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
Converse theorem: if the sum of the opposite angles of a quadrilateral is 180 0, then a circle can be described around it. Given: ABC D, A + C = 180 0 A B C D O Prove: Surround (O; R) is described around ABC D Proof: No. 729 (textbook) Which quadrilateral cannot be described around a circle?
Corollary 1: around any rectangle you can describe a circle, its center is the point of intersection of the diagonals. Corollary 2: a circle can be described around 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
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Slide captions:
8th grade L.S. Atanasyan Geometry 7-9 Inscribed and Circumscribed Circles
O D B C If all sides of a polygon touch a circle, then the circle is said to be inscribed in the polygon. A E A the polygon is said to be circumscribed 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 when studying the inscribed circle? A E O K Property of a tangent Property of tangent segments F P
D B C 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 equilateral 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. A O If the sums of the opposite sides of a convex quadrilateral are equal, then a circle can be inscribed in it. BC + A D = AB + DC
D B C Is it possible to inscribe a circle in this quadrilateral? A O 5 + 7 = 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 a triangle 2) C OL = CO M, along the hypotenuse and remainder. angle O L = M O Let us draw perpendiculars from point O to the sides of the triangle 3) MOA = KOA, along the hypotenuse and rest. corner MO = KO 4) L O= M O= K O point O is equidistant from the sides of the triangle. This means that a circle with center at t.O passes through points K, L and M. The sides of triangle ABC touch this circle. This means that the circle is an inscribed circle of 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 a circumscribed polygon is equal to 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
O D B C If all the vertices of a polygon lie on a circle, then the circle is called circumscribed about the polygon. A E A the 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 when studying the circumcircle? Inscribed angle theorem
O A B D In any cyclic quadrilateral, the sum of the opposite angles is 180 0. C + 360 0
59 0 ? 90 0 ? 65 0 ? 100 0 D А В С О 80 0 115 0 D А В С О 121 0 Find the unknown angles 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 described around 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 VO = CO 2) B OL = COL, along the legs 3) COM = A O M, along the legs CO = AO 4) VO=CO=AO, i.e. e. point O is equidistant from the vertices of the triangle. This means that a circle with a center at TO and radius OA will pass through all three vertices of the triangle, i.e. is a circumscribed circle.
K B C A A circle can be described around any triangle. L M Theorem O
O B C A O B C A No. 702 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 VSA 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 VSA 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 d i a m e t r
O VSA 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 equal to d and one of the acute angles of the triangle is equal to. d
O C V A No. 705 (a) A circle is circumscribed around 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 around a right triangle ABC with right angle C. Find the radius of this circle if AC=18 cm, 18 30 0 36 18 18
O B C A The lateral sides of the triangle shown in the figure are equal to 3 cm. Find the radius of the circle circumscribed around it. 180 0 3 3
O B C A The radius of the circle circumscribed about the triangle shown in the drawing is 2 cm. Find side AB. 180 0 2 2 45 0 ?
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