Central angle 2 times. Circle. Central and inscribed angle

Central corner is the angle whose vertex is at the center of the circle.
Inscribed angle An angle whose vertex lies on the circle and whose sides intersect it.

The figure shows central and inscribed angles, as well as their most important properties.

So, magnitude central corner equal to the angular value of the arc on which it rests. This means that a central angle of 90 degrees will be based on an arc equal to 90 °, that is, a circle. The central angle, equal to 60°, is based on an arc of 60 degrees, that is, on the sixth part of the circle.

The value of the inscribed angle is two times less than the central one based on the same arc.

Also, to solve problems, we need the concept of "chord".

Equal central angles are supported by equal chords.

1. What is the inscribed angle based on the diameter of the circle? Give your answer in degrees.

An inscribed angle based on a diameter is a right angle.

2. The central angle is 36° greater than the acute inscribed angle based on the same circular arc. Find the inscribed angle. Give your answer in degrees.

Let the central angle be x, and the inscribed angle based on the same arc be y.

We know that x = 2y.
Hence 2y = 36 + y,
y = 36.

3. The radius of the circle is 1. Find the value of an obtuse inscribed angle based on a chord equal to . Give your answer in degrees.

Let the chord AB be . An obtuse inscribed angle based on this chord will be denoted by α.
In triangle AOB, sides AO and OB are equal to 1, side AB is equal to . We have seen such triangles before. Obviously, the triangle AOB is right-angled and isosceles, that is, the angle AOB is 90 °.
Then the arc ASV is equal to 90°, and the arc AKB is equal to 360° - 90° = 270°.
The inscribed angle α rests on the AKB arc and is equal to half the angular value of this arc, i.e. 135°.

Answer: 135.

4. The chord AB divides the circle into two parts, the degree values ​​of which are related as 5:7. At what angle is this chord visible from point C, which belongs to the smaller arc of the circle? Give your answer in degrees.

The main thing in this task is the correct drawing and understanding of the condition. How do you understand the question: “At what angle is the chord visible from point C?”
Imagine that you are sitting at point C and you need to see everything that happens on chord AB. So, as if the chord AB is a screen in a cinema :-)
Obviously, you need to find the angle ACB.
The sum of the two arcs into which the chord AB divides the circle is 360°, i.e.
5x + 7x = 360°
Hence x = 30°, and then the inscribed angle ACB rests on an arc equal to 210°.
The value of the inscribed angle is equal to half the angular value of the arc on which it rests, which means that the angle ACB is equal to 105°.

Concept of inscribed and central angle

Let us first introduce the concept of a central angle.

Remark 1

Note that the degree measure of a central angle is equal to the degree measure of the arc it intercepts.

We now introduce the concept of an inscribed angle.

Definition 2

An angle whose vertex lies on a circle and whose sides intersect the same circle is called an inscribed angle (Fig. 2).

Figure 2. Inscribed angle

Inscribed angle theorem

Theorem 1

The measure of an inscribed angle is half the measure of the arc it intercepts.

Proof.

Let us be given a circle centered at the point $O$. Denote the inscribed angle $ACB$ (Fig. 2). The following three cases are possible:

• The ray $CO$ coincides with some side of the angle. Let this be the $CB$ side (Fig. 3).

Figure 3

In this case the arc $AB$ is less than $(180)^(()^\circ )$, hence the central angle $AOB$ is equal to the arc $AB$. Since $AO=OC=r$, the triangle $AOC$ is isosceles. Hence, the base angles $CAO$ and $ACO$ are equal. According to the theorem on the external angle of a triangle, we have:

• Beam $CO$ divides inner corner to two corners. Let it intersect the circle at the point $D$ (Fig. 4).

Figure 4

We get

• Ray $CO$ does not divide an interior angle into two angles and does not coincide with any of its sides (Fig. 5).

Figure 5

Consider separately the angles $ACD$ and $DCB$. By what was proved in item 1, we get

We get

The theorem has been proven.

Let's bring consequences from this theorem.

Corollary 1: The inscribed angles that intersect the same arc are equal.

Consequence 2: An inscribed angle that intercepts a diameter is a right angle.

Inscribed angle, problem theory. Friends! In this article, we will talk about tasks, for the solution of which it is necessary to know the properties of an inscribed angle. This is a whole group of tasks, they are included in the exam. Most of them are solved very simply, in one step.

There are more difficult tasks, but they will not present much difficulty for you, you need to know the properties of the inscribed angle. Gradually, we will analyze all the prototypes of tasks, I invite you to the blog!

Now the necessary theory. Recall what a central and inscribed angle, chord, arc, on which these angles rely:

The central angle in a circle is called a flat angle withpinnacle at its center.

The part of a circle that is inside a flat cornercalled an arc of a circle.

The degree measure of an arc of a circle is the degree measurecorresponding central angle.

An angle is called inscribed in a circle if the vertex of the angle lieson a circle, and the sides of the angle intersect this circle.

A line segment that connects two points on a circle is calledchord. The longest chord passes through the center of the circle and is calleddiameter.

To solve problems for angles inscribed in a circle,you need to know the following properties:

1. The inscribed angle is equal to half the central angle based on the same arc.

2. All inscribed angles based on the same arc are equal.

3. All inscribed angles based on the same chord, the vertices of which lie on the same side of this chord, are equal.

4. Any pair of angles based on the same chord, the vertices of which lie on opposite sides of the chord, add up to 180°.

Corollary: Opposite angles of a quadrilateral inscribed in a circle add up to 180 degrees.

5. All inscribed angles based on the diameter are straight.

In general, this property is a consequence of property (1), this is its special case. Look - the central angle is equal to 180 degrees (and this developed angle is nothing but a diameter), which means that according to the first property, the inscribed angle C is equal to its half, that is, 90 degrees.

Knowledge of this property helps in solving many problems and often allows you to avoid unnecessary calculations. Having mastered it well, you will be able to solve more than half of this type of problems orally. Two consequences that can be made:

Corollary 1: if a triangle is inscribed in a circle and one of its sides coincides with the diameter of this circle, then the triangle is right-angled (vertex right angle lies on the circle).

Corollary 2: the center of the described about right triangle circle coincides with the midpoint of its hypotenuse.

Many prototypes of stereometric problems are also solved by using this property and these corollaries. Remember the fact itself: if the diameter of a circle is a side of an inscribed triangle, then this triangle is right-angled (the angle opposite the diameter is 90 degrees). You can draw all the other conclusions and consequences yourself, you do not need to teach them.

As a rule, half of the problems for an inscribed angle are given with a sketch, but without notation. To understand the process of reasoning when solving problems (below in the article), the designations of vertices (corners) are introduced. On the exam, you can not do this.Consider the tasks:

What is an acute inscribed angle that intercepts a chord equal to the radius of the circle? Give your answer in degrees.

Let's build a central angle for a given inscribed angle, denote the vertices:

According to the property of an angle inscribed in a circle:

The angle AOB is equal to 60 0, since the triangle AOB is equilateral, and in an equilateral triangle all angles are equal to 60 0 . The sides of the triangle are equal, since the condition says that the chord is equal to the radius.

Thus, the inscribed angle DIA is 30 0 .

Answer: 30

Find the chord on which the angle 30 0 rests, inscribed in a circle of radius 3.

This is essentially the inverse problem (of the previous one). Let's build a central corner.

It is twice as large as the inscribed one, that is, the angle AOB is 60 0 . From this we can conclude that the triangle AOB is equilateral. Thus, the chord is equal to the radius, that is, three.

Answer: 3

The radius of the circle is 1. Find the value of an obtuse inscribed angle based on a chord equal to the root of two. Give your answer in degrees.

Let's build the central angle:

Knowing the radius and chord, we can find the central angle DIA. This can be done using the law of cosines. Knowing the central angle, we can easily find the inscribed angle ACB.

Cosine theorem: the square of any side of a triangle is equal to the sum squares of the other two sides, without doubling the product of these sides by the cosine of the angle between them.

Therefore, the second central angle is 360 0 – 90 0 = 270 0 .

According to the property of an inscribed angle, the angle DIA is equal to its half, that is, 135 degrees.

Answer: 135

Find the chord on which the angle of 120 degrees, the root of three, is inscribed in a circle of radius.

Connect points A and B with the center of the circle. Let's call it O:

We know the radius and inscribed angle DIA. We can find the central angle AOB (greater than 180 degrees), then find the angle AOB in triangle AOB. And then, using the cosine theorem, calculate AB.

By the property of an inscribed angle, the central angle AOB (which is greater than 180 degrees) will be equal to twice the inscribed angle, that is, 240 degrees. This means that the angle AOB in the triangle AOB is 360 0 - 240 0 = 120 0 .

According to the law of cosines:

Answer:3

Find the inscribed angle based on the arc that is 20% of the circle. Give your answer in degrees.

By the property of an inscribed angle, it is half the size of the central angle based on the same arc, in this case we are talking about the arc AB.

It is said that the arc AB is 20 percent of the circumference. This means that the central angle AOB is also 20 percent of 360 0 .* A circle is a 360 degree angle. Means,

Thus, the inscribed angle ACB is 36 degrees.

Answer: 36

arc of a circle AC, not containing points B, is 200 degrees. And the arc of the circle BC, which does not contain points A, is 80 degrees. Find the inscribed angle ACB. Give your answer in degrees.

Let us denote for clarity the arcs whose angular measures are given. Arc corresponding to 200 degrees - Blue colour, the arc corresponding to 80 degrees is red, the rest of the circle is yellow.

Thus, the degree measure of the arc AB (yellow), and hence the central angle AOB is: 360 0 – 200 0 – 80 0 = 80 0 .

The inscribed angle DAB is half the central angle AOB, that is, equal to 40 degrees.

Answer: 40

What is the inscribed angle based on the diameter of the circle? Give your answer in degrees.

In this article I will tell you how to solve problems that use .

First, as usual, let's recall the definitions and theorems that you need to know in order to successfully solve problems on .

1.Inscribed angle is an angle whose vertex lies on the circle and whose sides intersect the circle:

2.Central corner is the angle whose vertex coincides with the center of the circle:

Degree magnitude of the arc of a circle measured by the value of the central angle that it rests on.

In this case, the degree value of the AC arc is equal to the value of the angle AOC.

3. If the inscribed and central angles are based on the same arc, then the inscribed angle is twice the central angle:

4. All inscribed angles that lean on one arc are equal to each other:

5. The inscribed angle based on the diameter is 90°:

We will solve several problems.

1 . Task B7 (#27887)

Let's find the value of the central angle, which relies on the same arc:

Obviously, the value of the angle AOC is 90°, therefore, the angle ABC is 45°

Answer: 45°

2. Task B7 (No. 27888)

Find the angle ABC. Give your answer in degrees.

Obviously, the angle AOC is 270°, then the angle ABC is 135°.

Answer: 135°

3 . Task B7 (#27890)

Find the degree value of the arc AC of the circle on which the angle ABC rests. Give your answer in degrees.

Let's find the value of the central angle, which relies on the arc AC:

The value of the angle AOC is 45°, therefore, the degree measure of the arc AC is 45°.

Answer: 45°.

4 . Task B7 (#27885)

Find the angle ACB if the inscribed angles ADB and DAE are based on arcs of a circle, the degree values ​​of which are, respectively, and . Give your answer in degrees.

The angle ADB rests on the arc AB, therefore, the value of the central angle AOB is 118°, therefore, the angle BDA is 59°, and the adjacent angle ADC is 180°-59°=121°

Similarly, the angle DOE is 38° and the corresponding inscribed angle DAE is 19°.

Consider triangle ADC:

The sum of the angles of a triangle is 180°.

The value of the angle ASV is 180°- (121°+19°)=40°

Answer: 40°

5 . Task B7 (#27872)

The sides of the quadrilateral ABCD AB, BC, CD and AD subtend the arcs of the circumscribed circle, the degree values ​​of which are , , and , respectively. Find angle B of this quadrilateral. Give your answer in degrees.

Angle B rests on the arc ADC, the value of which is equal to the sum of the values ​​of the arcs AD and CD, i.e. 71°+145°=216°

The inscribed angle B is equal to half the value of the arc ADC, i.e. 108°

Answer: 108°

6. Task B7 (#27873)

Points A, B, C, D, located on a circle, divide this circle into four arcs AB, BC, CD and AD, the degree values ​​of which are related respectively as 4:2:3:6. Find angle A of quadrilateral ABCD. Give your answer in degrees.

(see the drawing of the previous task)

Since we have given the ratio of the magnitudes of the arcs, we introduce the unit element x. Then the magnitude of each arc will be expressed as follows:

AB=4x, BC=2x, CD=3x, AD=6x. All arcs form a circle, that is, their sum is 360 °.

4x+2x+3x+6x=360°, hence x=24°.

Angle A rests on the arcs BC and CD, which in total have a value of 5x=120°.

Therefore, angle A is 60°

Answer: 60°

7. Task B7 (#27874)

quadrilateral ABCD inscribed in a circle. Corner ABC equals , angle CAD

This is the angle formed by two chords originating at one point on the circle. An inscribed angle is said to be relies on an arc enclosed between its sides.

Inscribed angle equal to half of the arc on which it rests.

In other words, inscribed angle includes as many degrees, minutes and seconds as arc degrees, minutes and seconds are enclosed in half of the arc on which it relies. For justification, we analyze three cases:

First case:

Center O is located on the side inscribed angle ABS. Drawing the radius AO, we get ΔABO, in which OA = OB (as radii) and, accordingly, ∠ABO = ∠BAO. In relation to this triangle, the angle AOC is external. And so, it is equal to the sum of the angles ABO and BAO, or equal to the double angle ABO. So ∠ABO is half central corner AOC. But this angle is measured by arc AC. That is, the inscribed angle ABC is measured by half the arc AC.

Second case:

The center O is located between the sides inscribed angle ABC. Having drawn the diameter BD, we will divide the angle ABC into two angles, of which, according to the established in the first case, one is measured by half arcs AD, and the other half of the arc CD. And accordingly, the angle ABC is measured by (AD + DC) / 2, i.e. 1/2 AC.

Third case:

Center O is located outside inscribed angle ABS. Having drawn the diameter BD, we will have: ∠ABС = ∠ABD - ∠CBD . But the angles ABD and CBD are measured, based on the previously substantiated halves arcs AD and CD. And since ∠ABС is measured by (AD-CD)/2, that is, half of the AC arc.

Consequence 1. Any , based on the same arc are the same, that is, they are equal to each other. Since each of them is measured by half of the same arcs .

Consequence 2. Inscribed angle, based on the diameter - right angle. Since each such angle is measured by half a semicircle and, accordingly, contains 90 °.