Inscribed quadrilateral - a quadrilateral whose vertices all lie on the same circle.
Obviously, this circle will be called described around the quadrangle.
Described a quadrilateral is such that all its sides touch one circle. In this case the circle inscribed into a quadrangle.
The figure shows inscribed and circumscribed quadrilaterals and their properties.
Let's see how these properties are used in solving Unified State Examination problems.
1. Two angles of a quadrilateral inscribed in a circle are 82° and 58°. Find the largest remaining angle. Give your answer in degrees.
The sum of the opposite angles of an inscribed quadrilateral is 180°. Let angle A be 82°. Then there is an angle of 98 degrees opposite it. If angle B is 58°, then angle D is 180° - 58° = 122°.
Answer: 122.
2. The three sides of a quadrilateral circumscribed about a circle are in the ratio (in sequential order) as 1:2:3. Find the longest side of this quadrilateral if it is known that its perimeter is 32.
Let side AB be x, AD be 2x, and DC be 3x. According to the property of the described quadrilateral, the sums of opposite sides are equal, and therefore
x + 3x = BC + 2x.
It turns out that BC is equal to 2x. Then the perimeter of the quadrilateral is 8x. We get that x = 4 and the larger side is 12.
3. A trapezoid is described around a circle, the perimeter of which is 40. Find its midline.
We remember that midline trapezoid is equal to half the sum of the bases. Let the bases of the trapezoid be equal to a and c, and sides- b and d. According to the property of the described quadrilateral,
a + c = b + d, which means the perimeter is 2(a + c).
We get that a + c = 20, and the middle line is 10.
Let us once again repeat the properties of an inscribed and circumscribed quadrilateral.
A quadrilateral can be inscribed in a circle if and only if the sum of its opposite angles is equal to 180°.
A quadrilateral can be circumscribed around a circle if and only if the sums of the lengths of its opposite sides are equal.
Back Forward
Attention! Slide previews are for informational purposes only and may not represent all of the presentation's features. If you are interested this work, please download the full version.
Goals.
Educational. Creating conditions for successful mastery of the concept of the described quadrilateral, its properties, characteristics and mastering the skills to apply them in practice.
Developmental. Development of mathematical abilities, creation of conditions for the ability to generalize and apply forward and backward train of thought.
Educational. Cultivating a sense of beauty through the aesthetics of drawings, surprise at the unusual
decision, formation of organization, responsibility for the results of one’s work.
1. Study the definition of a circumscribed quadrilateral.
2. Prove the property of the sides of the circumscribed quadrilateral.
3. Introduce the duality of the properties of the sums of opposite sides and opposite angles of inscribed and circumscribed quadrilaterals.
4. To provide experience in the practical application of the considered theorems when solving problems.
5. Conduct initial monitoring of the level of assimilation of new material.
Equipment:
- computer, projector;
- textbook “Geometry. 10-11 grades” for general education. institutions: basic and profile. auto levels A.V. Pogorelov.
Software: Microsoft Word, Microsoft Power Point.
Using a computer when preparing a teacher for a lesson.
Using a standard Windows operating system program, the following were created for the lesson:
- Presentation.
- Tables.
- Drawings.
- Handout material.
Lesson Plan
Lesson progress
1. Organizational moment. Greetings. State the topic and purpose of the lesson. Record the date and topic of the lesson in your notebook.
2. Checking homework.
3. Studying new material.
Work on the concept of a circumscribed polygon.
Definition. The polygon is called described about a circle, if All his sides concern some circle.
Question. Which of the proposed polygons are described and which are not and why?
<Презентация. Слайд №2>
Proof of the properties of the circumscribed quadrilateral.
<Презентация. Слайд №3>
Theorem. In a circumscribed quadrilateral, the sums of opposite sides are equal.
Students work with a textbook and write down the formulation of the theorem in a notebook.
1. Present the formulation of the theorem in the form of a conditional sentence.
2. What is the condition of the theorem?
3. What is the conclusion of the theorem?
Answer. If a quadrilateral is circumscribed about a circle, That the sums of the opposite sides are equal.
The proof is carried out, students make notes in their notebooks.
<Презентация. Слайд №4>
Teacher. Note duality situations for sides and angles of circumscribed and inscribed quadrilaterals.
Consolidation of acquired knowledge.
Tasks.
Answer. 1. 10 m. 2. 20 m. 3. 21 m
Proof of the characteristic of a circumscribed quadrilateral.
State the converse theorem.
Answer. If in a quadrilateral the sums of opposite sides are equal, then a circle can be inscribed in it. (Return to slide 2, Fig. 7) <Презентация. Слайд №2>
Teacher. Clarify the formulation of the theorem.
Theorem. If the sums of opposite sides convex quadrilateral are equal, then a circle can be inscribed in it.
Working with the textbook. Get acquainted with the proof of the test for a circumscribed quadrilateral using the textbook.
Application of acquired knowledge.
3. Tasks based on finished drawings.
1. Is it possible to inscribe a circle in a quadrilateral with opposite sides 9 m and 4 m, 10 m and 3 m?
2. Is it possible to inscribe a circle into an isosceles trapezoid with bases of 1 m and 9 m, and a height of 3 m?
<Презентация. Слайд №6>
Written work in notebooks
.Task. Find the radius of a circle inscribed in a rhombus with diagonals 6 m and 8 m.
<Презентация. Слайд № 7>
4. Independent work.
1 option
1. Is it possible to inscribe a circle
1) into a rectangle with sides 7 m and 10 m,
2. The opposite sides of a quadrilateral circumscribed about a circle are 7 m and 10 m.
Find the perimeter of the quadrilateral.
3. An equilateral trapezoid with bases 4 m and 16 m is described around a circle.
1) radius of the inscribed circle,
Option 2
1. Is it possible to inscribe a circle:
1) in a parallelogram with sides 6 m and 13 m,
2) squared?
2. The opposite sides of a quadrilateral circumscribed about a circle are 9 m and 11 m. Find the perimeter of the quadrilateral.
3. An equilateral trapezoid with a side side of 5 m is described around a circle with a radius of 2 m.
1) the base of the trapezoid,
2) radius of the circumscribed circle.
5. Homework. P.86, No. 28, 29, 30.
6. Lesson summary. Independent work is checked and grades are given.
<Презентация. Слайд № 8>
The video course “Get an A” includes all the topics necessary for successful passing the Unified State Exam in mathematics for 60-65 points. Completely all tasks 1-13 of the Profile Unified State Exam in mathematics. Also suitable for passing the Basic Unified State Examination in mathematics. If you want to pass the Unified State Exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!
Preparation course for the Unified State Exam for grades 10-11, as well as for teachers. Everything you need to solve Part 1 of the Unified State Exam in mathematics (the first 12 problems) and Problem 13 (trigonometry). And this is more than 70 points on the Unified State Exam, and neither a 100-point student nor a humanities student can do without them.
All the necessary theory. Quick ways solutions, pitfalls and secrets of the Unified State Exam. All current tasks of part 1 from the FIPI Task Bank have been analyzed. The course fully complies with the requirements of the Unified State Exam 2018.
The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.
Hundreds of Unified State Exam tasks. Word problems and probability theory. Simple and easy to remember algorithms for solving problems. Geometry. Theory, reference material, analysis of all types of Unified State Examination tasks. Stereometry. Tricky solutions, useful cheat sheets, development of spatial imagination. Trigonometry from scratch to problem 13. Understanding instead of cramming. Clear explanations of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. Basis for solution complex tasks 2 parts of the Unified State Exam.
Theorem 1. The sum of the opposite angles of a cyclic quadrilateral is 180°.
Let a quadrilateral ABCD be inscribed in a circle with center O (Fig. 412). It is required to prove that ∠A + ∠C = 180° and ∠B + ∠D = 180°.
∠A, as inscribed in the circle O, measures 1 / 2 \(\breve(BCD)\).
∠C, as inscribed in the same circle, measures 1 / 2 \(\breve(BAD)\).
Consequently, the sum of angles A and C is measured by the half-sum of arcs BCD and BAD; in sum, these arcs make up a circle, i.e. have 360°.
Hence ∠A + ∠C = 360°: 2 = 180°.
It is similarly proven that ∠B + ∠D = 180°. However, this can be deduced in another way. We know that the sum of the interior angles of a convex quadrilateral is 360°. The sum of angles A and C is equal to 180°, which means that the sum of the other two angles of the quadrilateral also remains 180°.
Theorem 2 (converse). If in a quadrilateral the sum of two opposite angles is equal 180° , then a circle can be described around such a quadrilateral.
Let the sum of the opposite angles of the quadrilateral ABCD be equal to 180°, namely
∠A + ∠C = 180° and ∠B + ∠D = 180° (Fig. 412).
Let us prove that a circle can be described around such a quadrilateral.
Proof. Through any 3 vertices of this quadrilateral you can draw a circle, for example through points A, B and C. Where will point D be located?
Point D can only take one of the following three positions: be inside the circle, be outside the circle, be on the circumference of the circle.
Let’s assume that the vertex is inside the circle and takes position D’ (Fig. 413). Then in the quadrilateral ABCD’ we will have:
∠B + ∠D’ = 2 d.
Continuing side AD’ to the intersection with the circle at point E and connecting points E and C, we obtain the cyclic quadrilateral ABCE, in which, by the direct theorem
∠B + ∠E = 2 d.
From these two equalities it follows:
∠D’ = 2 d- ∠B;
∠E = 2 d- ∠B;
but this cannot be, since ∠D’, being external relative to the triangle CD’E, must be greater than angle E. Therefore, point D cannot be inside the circle.
It is also proved that vertex D cannot take position D" outside the circle (Fig. 414).
It remains to recognize that vertex D must lie on the circumference of the circle, i.e., coincide with point E, which means that a circle can be described around the quadrilateral ABCD.
Consequences.
1. A circle can be described around any rectangle.
2. A circle can be described around an isosceles trapezoid.
In both cases, the sum of opposite angles is 180°.
Theorem 3. In the circumscribed quadrilateral, the sums of opposite sides are equal. Let the quadrilateral ABCD be described about a circle (Fig. 415), that is, its sides AB, BC, CD and DA are tangent to this circle.
It is required to prove that AB + CD = AD + BC. Let us denote the points of tangency by the letters M, N, K, P. Based on the properties of tangents drawn to a circle from one point, we have:
Let us add these equalities term by term. We get:
AR + BP + DN + CN = AK + VM + DK + SM,
i.e. AB + CD = AD + BC, which is what needed to be proven.
Other materialsFor a triangle, both an inscribed circle and a circumcircle are always possible.
For a quadrilateral, a circle can be inscribed only if the sums of its opposite sides are the same. Of all the parallelograms, only a rhombus and a square can be inscribed with a circle. Its center lies at the intersection of the diagonals.
A circle can be described around a quadrilateral only if the sum of the opposite angles is 180°. Of all the parallelograms, only a rectangle and a square can be described as a circle. Its center lies at the intersection of the diagonals.
It is possible to describe a circle around a trapezoid, or a circle can be inscribed in a trapezoid if the trapezoid is isosceles.
Circumcenter
Theorem. The center of a circle circumscribed about a triangle is the point of intersection of the perpendicular bisectors to the sides of the triangle.
The center of a circle circumscribed about a polygon is the point of intersection of the perpendicular bisectors to the sides of this polygon.
Center Inscribed Circle
Definition. A circle inscribed in a convex polygon is a circle that touches all sides of this polygon (that is, each side of the polygon is tangent to the circle).
The center of the inscribed circle lies inside the polygon.
A polygon into which a circle is inscribed is called circumscribed.
A circle can be inscribed in a convex polygon if the bisectors of all its interior angles intersect at one point.
Center of a circle inscribed in a polygon- the point of intersection of its bisectors.
The center of the inscribed circle is equidistant from the sides of the polygon. The distance from the center to any side is equal to the radius of the inscribed circle. According to the property of tangents drawn from one point, any vertex of the circumscribed polygon is equidistant from the tangent points lying on the sides extending from this vertex.
A circle can be inscribed in any triangle. The center of a circle inscribed in a triangle is called the incenter.
A circle can be inscribed in a convex quadrilateral if and only if the sums of the lengths of its opposite sides are equal. In particular, a circle can be inscribed in a trapezoid if the sum of its bases is equal to the sum of its sides.
A circle can be inscribed in any regular polygon. You can also describe a circle around any regular polygon. The center of the incircle and circumcircle lie at the center of a regular polygon.
For any circumscribed polygon, the radius of the inscribed circle can be found using the formula
Where S is the area of the polygon, p is its semi-perimeter.
Regular n-gon - formulas
Formulas for the side length of a regular n-gon
1. Formula for the side of a regular n-gon in terms of the radius of the inscribed circle:
2. Formula for the side of a regular n-gon in terms of the radius of the circumscribed circle:
Formula for the incircle radius of a regular n-gon
Formula for the radius of the inscribed circle of an n-gon using the length of the side:
4. Circumcision radius formula regular triangle through side length:
6. Formula for the area of a regular triangle in terms of the radius of the inscribed circle: S = r 2 3√3
7. Formula for the area of a regular triangle in terms of the radius of the circumscribed circle:
4. Formula for the circumradius of a regular quadrilateral in terms of side length:
2. Formula for the side of a regular hexagon in terms of the circumradius: a = R
3. Formula for the radius of the inscribed circle of a regular hexagon in terms of the length of the side:
6. Formula for the area of a regular hexagon in terms of the radius of the inscribed circle: S = r 2 2√3
7. Formula for the area of a regular hexagon in terms of the radius of the circumscribed circle:
| S= | R 2 3√3 |
8. Angle between the sides of a regular hexagon: α = 120°
Number meaning(pronounced "pi") - mathematical constant, equal to the ratio
the circumference of a circle to the length of its diameter, it is expressed as an infinite decimal fraction.
Denoted by the letter "pi" of the Greek alphabet. What is pi equal to? IN simple cases It is enough to know the first 3 signs (3.14).
53. Find the length of the arc of a circle of radius R corresponding to the central angle of n°
The central angle subtended by an arc whose length is equal to the radius of the circle is called an angle of 1 radian.
The degree measure of an angle of 1 radian is:
Since the arc length π
R (semicircle), subtends central angle at 180 °
, then an arc of length R subtends the angle into π
times smaller, i.e. 
And vice versa
Because π = 3.14, then 1 rad = 57.3°
If the angle contains a radian, then it degree measure equal to 
And vice versa 
Usually, when denoting the measure of an angle in radians, the name “rad” is omitted.
For example, 360° = 2π rad, they write 360° = 2π
The table shows the most common angles in degrees and radians.