The angle formed by tangents to a circle. Secant and chords in a circle

Tangent to a circle. Dear friends! The base of tasks for the Unified State Exam in mathematics includes a group of problems where the condition deals with a tangent and raises the question of calculating an angle. These tasks are extremely simple. A little theory:

What is a tangent to a circle?

It is important to remember one basic property of a tangent:

In the presented problems, two more properties related to angles are used:

1. The sum of the angles of a quadrilateral is 360 0, more details.

2. The sum of the acute angles of a right triangle is 90 0.

Let's consider the tasks:

27879. Through the ends A And B arcs of a circle at 62 0 tangents are drawn A.C. And B.C.. Find the angle ACB. Give your answer in degrees.

It is said that the degree measure of arc AB corresponds to 62 degrees, that is, angle AOB is equal to 62 0 .

First way.

It is known that the sum of the angles in a quadrilateral is 360 0.

Second way.

In triangle ABC we can find angles ABC and BAC. Let's use the tangent property.

Since BC is a tangent, the angle OBC is equal to 90 0, which means:

Likewise

In an isosceles triangle AOB:

Means

According to the theorem on the sum of angles of a triangle:

Answer: 118 0

27880. Tangents C.A. And C.B. form an angle to the circle ACB, equal to 122 0. Find the magnitude of the minor arc AB, contracted by points of tangency. Give your answer in degrees.

The task is the opposite of the previous one. It is necessary to find the angle AOB.

Since BC and AC are tangent, then by the tangent property:

It is known that the sum of the angles in a quadrilateral is 360 0 .

In the quadrilateral OASV we know three angles, we can find the fourth:

Answer: 58

27882. Angle ACO is equal to 28 0, where O- center of the circle. His side C.A. touches the circle. Find the magnitude of the minor arc AB circle contained within this angle. Give your answer in degrees.

The degree value of the arc corresponds to the angle AOS. That is, the problem comes down to finding the angle AOC in the right triangle OCA. The triangle is rectangular because AC is a tangent, and the angle between the tangent and the radius drawn to the tangent point is 90 degrees.

According to the property of a right triangle, the sum of its acute angles is equal to 90 0, which means:

Answer: 62

27883. Find the angle ACO if his side C.A. touches the circle O- the center of the circle, and the major arc AD the circle contained within this angle is equal to 116 0. Give your answer in degrees.

It is said that the arc AD the circle enclosed inside the angle ASO is equal to 116 0, that is, the angle DOA is equal to 116 0. Triangle OCA is rectangular.

Angles AOC and DOA are adjacent, that is, their sum is equal to 180 0, which means:

The required angle is:

Answer: 26

Objective of the lesson: with formulate and prove the properties of another type of angles related to the concept of a circle - the angles between the tangent to the circle and the chord drawn to the point of tangency.

Lesson objectives:

• educational: test knowledge of theoretical material on the topic “Angles inscribed in a circle”; consider the connection between the degree measure of angles between a tangent and a chord with the degree measures of previously studied angles; practice problem solving skills using newly formulated properties;
• developing: development cognitive interest, curiosity, ability to analyze, observe and draw conclusions;

educational: increase interest in studying the subject of mathematics; fostering independence and activity.

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MOSCOW DEPARTMENT OF EDUCATION

STATE BUDGET EDUCATIONAL

INSTITUTION OF SECONDARY VOCATIONAL EDUCATION

COLLEGE OF LANDSCAPE DESIGN №18

Geometry lesson notes

9th grade

“Angles between a tangent to a circle and a chord drawn to the point of tangency”

Prepared

teacher of mathematics and computer science

Kolozyan Elina Shavarshevna

Moscow, 2012

Subject: Angles between a tangent to a circle and a chord drawn to a point

Touches

Objective of the lesson: with formulate and prove the properties of another type of angles related to the concept of a circle - the angles between the tangent to the circle and the chord drawn to the point of tangency.

Lesson objectives:

educational:test knowledge of theoretical material on the topic “Angles inscribed in a circle”; consider the connection between the degree measure of angles between a tangent and a chord with the degree measures of previously studied angles; practice problem solving skills using newly formulated properties;

developing: development of cognitive interest, curiosity, ability to analyze, observe and draw conclusions;

educational: increase interest in studying the subject of mathematics; fostering independence and activity.

During the classes

I. Oral work (according to Figure 1)

Oral work is carried out in order to orient students to independent work, which will follow after this. The drawing that was used during the survey will be a hint, so in a strong class it can be removed, and in a weak class, on the contrary, it can be left.

U. What angles associated with a circle are you already familiar with? Give

Define and name them on the drawing

D.1) Central angle (<АОС), вершина которого находится в центре

Circles.

2) Inscribed in a circle (<АВС), его вершина лежит на окружности, стороны пересекают её.

U. How are the degree measures of these angles related?

D. The degree measure of an inscribed angle is equal to half its degree measure

The corresponding central angle (<АВС= <АОС).

U. How are their degree measures related to the arc on which they rest?

D.<АВС= ᵕ АС, <АОС= ᵕ АС.

U. What corollaries from the theorem about an angle inscribed in a circle have you already

Studied?

D. An angle inscribed in a circle and subtended by a diameter is a right angle.

Angles inscribed in a circle that rest on the same arc are equal.

II. Independent work(based on material discussed in oral work)

Independent work is aimed at testing knowledge of theoretical material. The first task is very simple, but only for those students who understand the connection between these concepts and do not memorize the formulations. This work will provide an opportunity to analyze the class’s perception of theoretical material. The second task is aimed at checking students’ independent work at home, since these consequences were discussed in class only orally, and written evidence was offered as homework. A grade of “3” in this work can be given for completing the first task and writing the correct formulation of the corollary in the second.

Option 1.

An angle inscribed in a circle is always ……………….of the corresponding central angle.

An angle inscribed in a circle always……………corresponds to the arc.

An arc of a circle always…………….corresponding inscribed angle.

The degree measure of an arc is always…………the corresponding central angle.

II. Formulate and prove the property of an angle inscribed in a circle that is supported by a diameter.

Option2.

I. Instead of the ellipsis, insert the correct answer:

2 times more; 2 times less; equals.

The degree measure of an arc is always ……………….to the corresponding central angle.

The central angle is always……………….corresponding to the arc.

An arc of a circle always……………corresponding inscribed angle.

The central angle is always……………….the corresponding inscribed angle.

An angle inscribed in a circle is always…………….of the corresponding arc.

An angle inscribed in a circle always…………corresponding to the central angle.

II. Formulate and prove the property of angles inscribed in a circle and supported by an arc.

Answers:

III. New material

The explanation of new material begins not with a proof, but with an oral problem, which leads students to independently formulate this property, and also facilitates understanding of the proof, since it repeats the stages of solving the problem.

1. Oral work based on a drawing on the board (Fig. 2)

Fig.2

U. Name the central angle in the drawing.

D.<АОВ - вершина угла в центре окружности.

U. What is called a chord?

D. A segment connecting two points on a circle; in our case AB.

U. Name the tangent to the circle. What property does it have?

D. Direct sun. The tangent is perpendicular to the radius drawn to the tangent point, which means<ОВС=90°.

The teacher marks this angle in the drawing.

U. Show the angles between the tangent and the chord drawn to the point of tangency. Select and label the smallest one.

D.<АВС=60° (90°-30°)

U.Name the arc contained between the tangent and the chord.

D. ᵕ AB

U. What angle is it equal to?

D. ᵕ AB=<АОВ (градусная мера дуги равна градусной мере соответствующего центрального угла).

Students write this wording under the drawing.

U. Calculate the degree measure of this angle.

D. AO=OB (radii), therefore, triangle AOB is isosceles with base AB, therefore,<А=<В=30°, следовательно <АОВ=180°-2*30° = 120°

U. Compare the degree measure of the angle between the tangent and the chord and the degree measure of the arc enclosed between the tangent and the chord.

D. The angle between the tangent and the chord drawn to the point of contact is equal to half the arc enclosed between them.

U. Guys, we have now formulated the property of the angle formed by a tangent to a circle and a chord drawn to the point of contact. Let's write this property down in our notebook.

Students take notes.

U. Why can’t we say that we have already proven this property?

D. a numerical example is not a proof, since we cannot go through all the numbers.

2. Written proof of the theorem

The teacher proves the theorem at the blackboard, the children write the proof in their notebooks.

THEOREM: The angle between the tangent and the chord drawn to the point of contact is equal to half the arc enclosed between them.

The proof of the theorem is based on an already solved problem; Students already explain the points that they have understood.

Fig.3

Given: Circle (O;r), MN - tangent, AB - chord, AB ∩MN = (A) (Fig. 3).

Prove:<ВАМ= ᵕ ВА.

Proof:

1. Additional construction: VO = AO (radii)

2. <АОМ=90°, так как MN - касательная, ОА- радиус, <ВАМ=90°- <ОАВ.

3. Consider the triangle BOA: OB = OA, which means the triangle is isosceles with base AB, therefore<ОАВ=<АВО.

<ВОА=180°- <ОАВ - <АВО=180°- 2*<ОАВ= 2*(90°-<ОАВ)

4.ᵕ VA=<ВОА=2*(90°-<ОАВ)= 2*<ВАМ, значит,

ᵕ VA=2*<ВАМ и <ВАМ= ᵕ ВА.

IV. Consolidation

When reinforcing new material, problems not from the textbook are used, so students are given printouts containing the tasks.

Tasks No. 1 and 2 are completed orally, No. 3,4 (optional) - in writing.

No. 1 (Fig. 4)

<АВС -?

Fig.4

Solution:

1. <АВС= ᵕ VA (property of the angle between a tangent and a chord).

ᵕ VA=<АОВ=180° (развернутый угол).

<АВС= *180°=90°.

No. 2 (Fig. 5)

<СВЕ-?

50°

Fig.5

Solution:

<СВЕ= ᵕ BC (property of the angle between a tangent and a chord).

<ВАС- вписанный окружность, значит <ВАС= ᵕ YOU (ᵕ BC) (property of an inscribed angle).

ᵕ BC= 2*<ВАС= 2*50°=100°, <СВЕ=100°:2=50°

No. 3. (Fig.6)

Fig.6

Solution:

ᵕ BEA=2*<АМВ (вписанный угол в 2 раза меньше дуги, на которую он опирается), следовательно, ᵕ BEA=2*80°=160°.

AEB=160°:2=80° (property between tangent and chord).

Consider triangle ADB:

Problems No. 2 and No. 3 are specifically considered in detail (angles are found through performing inverse operations: multiplying by 2, then dividing by 2). If none of the students notice the irrationality in the solution, it is necessary to focus the children’s attention on points 1.2 of task No. 3.

After this, you can formulate and write it as a property:

The angle between the tangent and the chord drawn to the point of tangency is equal to the inscribed angle subtended by the arc contained between the tangent and the chord.

No. 4. (Fig.7)

Given: triangle ABC is inscribed in a circle,<А:<В:<С=4:5:6;

VM - tangent to the circle.

Calculate:<МВС и <МВА.

Fig.7

Solution:

Consider triangle ABC:<А+<В+<С=180°.

Let x be the proportionality coefficient:

4x+5x+6x=180,

15x=180,

x=12.

<А=4*12°=48°, <МВС=<А=48° (свойство угла между касательной и хордой и вписанного угла, опирающегося на дугу, заключенную между касательной и хордой).

<АВМ=<АВС+<МВС=5*12°+48°=60°+48°=108°.

V. Lesson summary (work according to Fig. 8)

U. Name all the resulting inscribed angles.

D.<САВ, <АВС, <ВСА.

U.Name all the angles between the tangent and the chords.

D.

U.Which of them will be equal and why?

D.

U.Which angle of the triangle is equal to each of these three pairs and why?

D.

U. What can be said about the type of triangles ANB; BKC; CMA?

D. they are isosceles, since each of these triangles has two equal angles

VI. Homework

Learn theory (test preparation)

№ 54,59

Oral geometry, grades 7-9

Ershova A.P.

"Ilexa"

2004

Mathematical dictations

Geometry 7-11 grades

Levitas G.G.

"Ilexa"

2008

Berezina L.Yu.

"Exam"

Geometry lesson in 10th grade UMK L.S. Atanasyan

MBOU Verkhlichskaya secondary school, Krasnogorsk district, Bryansk region

Teacher: Strugovets Elena Vasilievna

Lesson topic:Angle between tangent and chord.

The purpose of the lesson:

Systematize students’ knowledge in the section of planimetry “Angles associated with a circle.” Prove the theorem about the angle between a tangent and a chord. Create meaningful and organizational conditions for schoolchildren to use a complex of knowledge to solve problems.

Develop students’ personal and semantic relationships to the subject being studied. To promote the formation of collective and independent work, to develop the ability to express one’s thoughts clearly and clearly.

To instill in students an interest in the subject through joint creative work; develop the ability to accurately and competently perform geometric constructions and mathematical notations.

Equipment:

Thematic tables.

Tests and answer cards.

During the classes.

Organizing time. (1 min)

Check students' readiness for the lesson and mark those who are absent.

Setting a goal. (2 minutes)

In your notebook, write down the date and topic of the lesson. In the lesson we will review theoretical knowledge on the topic “Angles associated with a circle.” Let's prove the theorem about the angle between a tangent and a chord, and learn how to apply it to solving problems of various types.

Updating knowledge. (7 min)

Dictation (followed by testing). Finish the sentence you read.

An angle whose vertex lies on a circle is called... (inscribed).

An angle with a vertex in the center of a circle is ... (central).

A segment connecting two points on a circle is called... (chord).

The largest of the chords of circles is ... (diameter).

The measure of the arc is equal to the measure of ... (central angle).

A straight line that has only one common point with a circle is called... (tangent)

The tangent to the circle and the radius drawn to the point of contact are mutually... (perpendicular)

A straight line that has two common points with a circle is called... (secant).

All inscribed angles based on the diameter ... (right)

An angle formed by two tangents drawn from one common point is called ... (circumscribed).

2) Solving problems according to the drawing.

3) Problem solving

The central angle AOB is 30 0 greater than the inscribed angle subtended by the arc AB. Find each of these angles.

Answer.30 0 ; 60 0 .

Answer.50 0 .

IV . Proof of the theorem.(5 minutes)

We know that an inscribed angle is measured by half the arc on which it rests. Let us prove the theorem about the angle between the tangent and the chord.

Theorem.
The angle between the tangent and the chord passing through the point of tangency is measured by half the arc contained in it.
Proof.

Fig.1

Let AB- given chord, SS 1 - tangent passing through a point A. If AB- diameter (Fig. 1), then enclosed inside the angle YOU(and also
angle YOU 1 ) an arc is a semicircle. On the other hand, angles YOU And YOU 1 in this case they are straight, so the theorem is true.

Fig.2
Let now the chordAB is not a diameter. For definiteness, we will assume that the pointsWITH And WITH 1 on the tangent are chosen so that the angleSAV-
sharp, and denote by the letter a the size of the arc contained in it (Fig. 2). Let's draw the diameterA D and note that the triangleAB D rectangular, soA D IN= 90° - D AB = YOU, Because the angle ABB inscribed, then A D IN= , and therefore YOU= . So the angle YOU between tangentsAC and chord AB measured by half the arc contained in it.
A similar statement is true for the angleYOU 1 . indeed, the cornersYOU And YOU 1 - adjacent, thereforeYOU 1 = 180-=. On the other hand, (360° - ) is the magnitude of the arcA D IN, enclosed inside the cornerYOU 1 . The theorem has been proven.

2. If

VI. Solving design problems. (7min)

1. Through a point D , lying on the radiusOA circle with centerABOUT , a chord is drawnSun , perpendicular toOA, and through the point IN a tangent to the circle is drawn intersecting straight line OA at pointE . Prove that the rayVA- bisector.

Proof.

ABE=AB – according to theoremabout the angle between the tangent and the chord. 4”

“3”

“2”

I know the definitions of types of angles

I can find angles when solving problems

Theorem on the angle between a tangent and a chord.

The proof of the theorem is clear

I apply the theorem to solve problems

Geometry lesson in 10th grade UMK L.S. Atanasyan

MBOU Verkhlichskaya secondary school, Krasnogorsk district, Bryansk region

Teacher: Strugovets Elena Vasilievna

Lesson topic:Angle between tangent and chord.

The purpose of the lesson:Prove the theorem about the angle between a tangent and a chord. Help students develop the ability to apply the learned theorem when solving problems.

Tasks:

Systematize students’ knowledge in the section of planimetry “Angles associated with a circle” Create meaningful and organizational conditions for schoolchildren to use a complex of knowledge to solve problems.

Develop students’ personal and semantic relationships to the subject being studied. To promote the formation of collective and independent work, to develop the ability to express one’s thoughts clearly and clearly.

To instill in students an interest in the subject through joint creative work; develop the ability to accurately and competently perform geometric constructions and mathematical notations.

Equipment:

Thematic tables, presentation.

Tests and answer cards.

During the classes.

Organizing time. (1 min)

Check students' readiness for the lesson and mark those who are absent.

Setting a goal. (2 minutes)

In your notebook, write down the date and topic of the lesson. In the lesson we will review theoretical knowledge on the topic “Angles associated with a circle.” Let's prove the theorem about the angle between a tangent and a chord, and learn how to apply it to solving problems of various types.

Updating knowledge. (7 min)

Dictation (followed by testing). Finish the sentence you read.

An angle whose vertex lies on a circle is called... (inscribed).

An angle with a vertex in the center of a circle is ... (central).

A segment connecting two points on a circle is called... (chord).

The largest of the chords of circles is ... (diameter).

The measure of the arc is equal to the measure of ... (central angle).

A straight line that has only one common point with a circle is called... (tangent)

The tangent to the circle and the radius drawn to the point of contact are mutually... (perpendicular)

A straight line that has two common points with a circle is called... (secant).

All inscribed angles based on the diameter ... (right)

An angle formed by two tangents drawn from one common point is called ... (circumscribed).

2) Solving problems according to the drawing.

3) Problem solving

The central angle AOB is 30 0 greater than the inscribed angle subtended by the arc AB. Find each of these angles.

Answer.30 0 ; 60 0 .

Answer.50 0 .

IV . Proof of the theorem.(5 minutes)

We know that an inscribed angle is measured by half the arc on which it rests. Let us prove the theorem about the angle between the tangent and the chord.

Theorem.
The angle between the tangent and the chord passing through the point of tangency is measured by half the arc contained in it.
Proof.

Fig.1

Let AB- given chord, SS 1 - tangent passing through a point A. If AB- diameter (Fig. 1), then enclosed inside the angle YOU(and also
angle YOU 1 ) an arc is a semicircle. On the other hand, angles YOU And YOU 1 in this case they are straight, so the theorem is true.

Fig.2
Let now the chordAB is not a diameter. For definiteness, we will assume that the pointsWITH And WITH 1 on the tangent are chosen so that the angleSAV-
sharp, and denote by the letter a the size of the arc contained in it (Fig. 2). Let's draw the diameterA D and note that the triangleAB D rectangular, soA D IN= 90° - D AB = YOU, Because the angle ABB inscribed, then A D IN= , and therefore YOU= . So the angle YOU between tangentsAC and chord AB measured by half the arc contained in it.
A similar statement is true for the angleYOU 1 . indeed, the cornersYOU And YOU 1 - adjacent, thereforeYOU 1 = 180-=. On the other hand, (360° - ) is the magnitude of the arcA D IN, enclosed inside the cornerYOU 1 . The theorem has been proven.

Solving problems using drawings. (5 minutes)

1. If

2. If

VI. Solving design problems. (7min)

1. Through a point D , lying on the radiusOA circle with centerABOUT , a chord is drawnSun , perpendicular toOA, and through the point IN a tangent to the circle is drawn intersecting straight line OA at pointE . Prove that the rayVA- bisector.

Proof.

ABE=AB – according to theoremabout the angle between the tangent and the chord.

ABC=AC – inscribed angle.

AB=AC – equal chords subtend equal arcs, and chords AB and AC are equal, since ABC is isosceles. Therefore, ABE = ABC, beamVA- bisector.

VII. Homework. ( 3 min)

1. In triangle ABC A=32 0, and C=24 0 . A circle with center at point B passes through point A, intersects AC at point M, BC at pointN. What is A equal to? N M?

2. Be able to prove a theorem.

VIII. Summarizing. Self-analysis of the lesson. (3 min)

Analysis of students' work in class. Making marks.

Self-analysis based on acquired knowledge

Student name: _______________________________________