Circle and inscribed angle. Visual guide (2019)

Basic terms.

How well do you remember all the names associated with the circle? Just in case, let us remind you - look at the pictures - refresh your knowledge.

Well, first of all - The center of a circle is a point from which the distances from all points on the circle are the same.

Secondly - radius - a line segment connecting the center and a point on the circle.

There are a lot of radii (as many as there are points on the circle), but All radii have the same length.

Sometimes for short radius they call it exactly length of the segment“the center is a point on the circle,” and not the segment itself.

And here's what happens if you connect two points on a circle? Also a segment?

So, this segment is called "chord".

Just as in the case of radius, diameter is often the length of a segment connecting two points on a circle and passing through the center. By the way, how are diameter and radius related? Look carefully. Of course radius equal to half diameter

In addition to chords, there are also secants.

Remember the simplest thing?

Central angle is the angle between two radii.

And now - the inscribed angle

Inscribed angle - the angle between two chords that intersect at a point on a circle.

In this case, they say that the inscribed angle rests on an arc (or on a chord).

Look at the picture:

Measurements of arcs and angles.

Circumference. Arcs and angles are measured in degrees and radians. First, about degrees. There are no problems for angles - you need to learn how to measure the arc in degrees.

The degree measure (arc size) is the value (in degrees) of the corresponding central angle

What does the word “appropriate” mean here? Let's look carefully:

Do you see two arcs and two central angles? Well, it corresponds to a larger arc larger angle(and it’s okay that it’s larger), and a smaller arc corresponds to a smaller angle.

So, we agreed: the arc contains the same number of degrees as the corresponding central angle.

And now about the scary thing - about radians!

What kind of beast is this “radian”?

Imagine: Radians are a way of measuring angles... in radii!

An angle measuring radians is like this central angle, the arc length of which is equal to the radius of the circle.

Then the question arises - how many radians are there in a straight angle?

In other words: how many radii “fit” in half a circle? Or in another way: how many times is the length of half a circle? greater than radius?

Scientists asked this question back in Ancient Greece.

And so, after a long search, they discovered that the ratio of the circumference to the radius does not want to be expressed in “human” numbers like, etc.

And it’s not even possible to express this attitude through roots. That is, it turns out that it is impossible to say that half a circle is times or times larger than the radius! Can you imagine how amazing it was for people to discover this for the first time?! For the ratio of the length of half a circle to the radius, “normal” numbers were not enough. I had to enter a letter.

So, - this is a number expressing the ratio of the length of the semicircle to the radius.

Now we can answer the question: how many radians are there in a straight angle? It contains radians. Precisely because half the circle is times larger than the radius.

Ancient (and not so ancient) people throughout the centuries (!) tried to calculate it more accurately mysterious number, it is better to express it (at least approximately) through “ordinary” numbers. And now we are incredibly lazy - two signs after a busy day are enough for us, we are used to

Think about it, this means, for example, that the length of a circle with a radius of one is approximately equal, but this exact length is simply impossible to write down with a “human” number - you need a letter. And then this circumference will be equal. And of course, the circumference of the radius is equal.

Let's go back to radians.

We have already found out that a straight angle contains radians.

What we have:

That means I'm glad, that is, I'm glad. In the same way, a plate with the most popular angles is obtained.

The relationship between the values of the inscribed and central angles.

There is an amazing fact:

The inscribed angle is half the size of the corresponding central angle.

Look how this statement looks in the picture. A “corresponding” central angle is one whose ends coincide with the ends of the inscribed angle, and the vertex is at the center. And at the same time, the “corresponding” central angle must “look” at the same chord () as the inscribed angle.

Why is this so? Let's figure it out first simple case. Let one of the chords pass through the center. It happens like that sometimes, right?

What happens here? Let's consider. It is isosceles - after all, and - radii. So, (labeled them).

Now let's look at. This is the outer corner for! Remember that the outer corner equal to the sums two internal ones, not adjacent to it, and write:

That is! Unexpected effect. But there is also a central angle for the inscribed.

This means that for this case they proved that the central angle is twice the inscribed angle. But it hurts too much special case: Isn’t it true that the chord doesn’t always go straight through the center? But it’s okay, now this particular case will help us a lot. Look: second case: let the center lie inside.

Let's do this: draw the diameter. And then... we see two pictures that were already analyzed in the first case. Therefore we already have that

This means (in the drawing, a)

Well, I stayed last case: center outside the corner.

We do the same thing: draw the diameter through the point. Everything is the same, but instead of a sum there is a difference.

That's it!

Let's now form two main and very important consequences from the statement that the inscribed angle is half the central angle.

Corollary 1

All inscribed angles based on one arc are equal to each other.

We illustrate:

There are countless inscribed angles based on the same arc (we have this arc), they may look completely different, but they all have the same central angle (), which means that all these inscribed angles are equal among themselves.

Corollary 2

The angle subtended by the diameter is a right angle.

Look: what angle is central to?

Certainly, . But he is equal! Well, therefore (as well as many more inscribed angles resting on) and is equal.

Angle between two chords and secants

But what if the angle we are interested in is NOT inscribed and NOT central, but, for example, like this:

or like this?

Is it possible to somehow express it through some central angles? It turns out that it is possible. Look: we are interested.

a) (as an external corner for). But - inscribed, rests on the arc -. - inscribed, rests on the arc - .

For beauty they say:

The angle between the chords is equal to half the sum of the angular values of the arcs enclosed in this angle.

They write this for brevity, but of course, when using this formula you need to keep in mind the central angles

b) And now - “outside”! How can this be? Yes, almost the same! Only now (again we apply the property of the external angle for). That is now.

And that means... Let’s bring beauty and brevity to the notes and wording:

The angle between the secants is equal to half the difference in the angular values of the arcs enclosed in this angle.

Well, now you are armed with all the basic knowledge about angles related to a circle. Go ahead, take on the challenges!

CIRCLE AND INSINALED ANGLE. MIDDLE LEVEL

Even a five-year-old child knows what a circle is, right? Mathematicians, as always, have an abstruse definition on this subject, but we will not give it (see), but rather let us remember what the points, lines and angles associated with a circle are called.

Important Terms

Well, first of all:

center of the circle- a point from which all points on the circle are the same distance.

Secondly:

There is another accepted expression: “the chord contracts the arc.” Here in the figure, for example, the chord subtends the arc. And if a chord suddenly passes through the center, then it has special name: "diameter".

By the way, how are diameter and radius related? Look carefully. Of course

And now - the names for the corners.

Natural, isn't it? The sides of the angle extend from the center - which means the angle is central.

This is where difficulties sometimes arise. Pay attention - NOT ANY angle inside a circle is inscribed, but only one whose vertex “sits” on the circle itself.

Let's see the difference in the pictures:

Another way they say:

There is one tricky point here. What is the “corresponding” or “own” central angle? Just an angle with the vertex at the center of the circle and the ends at the ends of the arc? Not really. Look at the drawing.

One of them, however, doesn’t even look like a corner - it’s bigger. But a triangle cannot have more angles, but a circle may well! So: the smaller arc AB corresponds to a smaller angle (orange), and the larger arc corresponds to a larger one. Just like that, isn't it?

The relationship between the magnitudes of the inscribed and central angles

Remember this very important statement:

In textbooks they like to write this same fact like this:

Isn’t it true that the formulation is simpler with a central angle?

But still, let’s find a correspondence between the two formulations, and at the same time learn to find in the drawings the “corresponding” central angle and the arc on which the inscribed angle “rests”.

Look: here is a circle and an inscribed angle:

Where is its “corresponding” central angle?

Let's look again:

What is the rule?

But! In this case, it is important that the inscribed and central angles “look” at the arc from one side. Here, for example:

Oddly enough, blue! Because the arc is long, longer than half the circle! So don’t ever get confused!

What consequence can be deduced from the “halfness” of the inscribed angle?

But, for example:

Angle subtended by diameter

You have already noticed that mathematicians love to talk about the same things. in different words? Why do they need this? You see, the language of mathematics, although formal, is alive, and therefore, as in ordinary language, every time I want to say it in a way that is more convenient. Well, we have already seen what “an angle rests on an arc” means. And imagine, the same picture is called “an angle rests on a chord.” Which one? Yes, of course, to the one that tightens this arc!

When is it more convenient to rely on a chord than on an arc?

Well, in particular, when this chord is a diameter.

There is a surprisingly simple, beautiful and useful statement for such a situation!

Look: here is the circle, the diameter and the angle that rests on it.

CIRCLE AND INSINALED ANGLE. BRIEFLY ABOUT THE MAIN THINGS

1. Basic concepts.

3. Measurements of arcs and angles.

An angle of radians is a central angle whose arc length is equal to the radius of the circle.

This is a number that expresses the ratio of the length of a semicircle to its radius.

The circumference of the radius is equal to.

4. The relationship between the values of the inscribed and central angles.

The concept of inscribed and central angle

Let us first introduce the concept of a central angle.

Note 1

Note that the degree measure of a central angle is equal to the degree measure of the arc on which it rests.

Let us 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 degree measure of an inscribed angle is equal to half degree measure the arc on which it rests.

Proof.

Let us be given a circle with center at point $O$. Let's denote the inscribed angle $ACB$ (Fig. 2). The following three cases are possible:

Ray $CO$ coincides with any side of the angle. Let this be the side $CB$ (Fig. 3).

Figure 3.

In this case, the arc $AB$ is less than $(180)^(()^\circ )$, therefore the central angle $AOB$ is equal to the arc $AB$. Since $AO=OC=r$, then the triangle $AOC$ is isosceles. This means that the base angles $CAO$ and $ACO$ are equal to each other. By the theorem about external angle triangle, we have:

Beam $CO$ divides internal corner at two angles. Let it intersect the circle at point $D$ (Fig. 4).

Figure 4.

We get

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

Figure 5.

Let us consider angles $ACD$ and $DCB$ separately. According to what was proved in point 1, we obtain

We get

The theorem is proven.

Let's give consequences from this theorem.

Corollary 1: Inscribed angles that rest on the same arc are equal to each other.

Corollary 2: An inscribed angle that subtends a diameter is a right angle.

Central angle is an angle whose vertex is at the center of the circle.
Inscribed angle- an angle whose vertex lies on a circle and whose sides intersect it.

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

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

The magnitude of the inscribed angle is two times smaller than the central angle based on the same arc.

Also, to solve problems we will need the concept of “chord”.

Equal central angles subtend equal chords.

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

An inscribed angle subtended by a diameter is a right angle.

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

Let the central angle be equal to x, and the inscribed angle subtended by the same arc be equal to y.

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

3. The radius of the circle is equal to 1. Find the value of the obtuse inscribed angle subtended by the chord, equal to . Give your answer in degrees.

Let the chord AB be equal to . The 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 already encountered such triangles. Obviously, triangle AOB is rectangular and isosceles, that is, angle AOB is 90°.
Then the arc ACB is equal to 90°, and the arc AKB is equal to 360° - 90° = 270°.
The inscribed angle α is based on the arc AKB and is equal to half angular magnitude of this arc, that is, 135°.

Answer: 135.

4. The chord AB divides the circle into two parts, the degree values of which are in the ratio 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 conditions. 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 is happening on the chord AB. It’s as if the chord AB is a screen in a movie theater :-)
Obviously, you need to find the angle ACB.
The sum of the two arcs into which the chord AB divides the circle is equal to 360°, that is
5x + 7x = 360°
Hence x = 30°, and then the inscribed angle ACB rests on an arc equal to 210°.
The magnitude of the inscribed angle is equal to half the angular magnitude of the arc on which it rests, which means that angle ACB is equal to 105°.

Planimetry is a branch of geometry that studies the properties flat figures. These include not only everyone famous triangles, squares, rectangles, but also straight lines and angles. In planimetry, there are also such concepts as angles in a circle: central and inscribed. But what do they mean?

What is a central angle?

In order to understand what a central angle is, you need to define a circle. A circle is the collection of all points equidistant from a given point (the center of the circle).

It is very important to distinguish it from a circle. You need to remember that a circle is a closed line, and a circle is a part of a plane bounded by it. A polygon or an angle can be inscribed in a circle.

A central angle is an angle whose vertex coincides with the center of the circle and whose sides intersect the circle at two points. The arc that an angle limits by its points of intersection is called the arc on which the given angle rests.

Let's look at example No. 1.

In the picture, angle AOB is central, because the vertex of the angle and the center of the circle are one point O. It rests on the arc AB, which does not contain point C.

How does an inscribed angle differ from a central angle?

However, in addition to central angles, there are also inscribed angles. What is their difference? Just like the central angle, the angle inscribed in the circle rests on a certain arc. But its vertex does not coincide with the center of the circle, but lies on it.

Let's give next example.

Angle ACB is called an angle inscribed in a circle with a center at point O. Point C belongs to the circle, that is, it lies on it. The angle rests on the arc AB.

In order to successfully cope with geometry problems, it is not enough to be able to distinguish between inscribed and central angles. As a rule, to solve them you need to know exactly how to find the central angle in a circle and be able to calculate its value in degrees.

So, the central angle is equal to the degree measure of the arc on which it rests.

In the picture, angle AOB rests on arc AB equal to 66°. This means that angle AOB is also 66°.

Thus, the central angles subtended by equal arcs, are equal.

In the figure, arc DC is equal to arc AB. So angle AOB equal to angle DOC.

It may seem that the angle inscribed in the circle is equal to the central angle, which rests on the same arc. However, this is a grave mistake. In fact, even just looking at the drawing and comparing these angles with each other, you can see that their degree measures will have different meanings. So what is the inscribed angle in a circle?

The degree measure of an inscribed angle is equal to one-half of the arc on which it rests, or half the central angle if they rest on the same arc.

Let's look at an example. Angle ASV rests on an arc equal to 66°.

This means angle ACB = 66°: 2 = 33°

Let's consider some consequences from this theorem.

Inscribed angles, if they are based on the same arc, chord, or equal arcs, are equal.
If inscribed angles rest on one chord, but their vertices lie along different sides from it, the sum of the degree measures of such angles is 180°, since in this case both angles rest on arcs, the degree measure of which in total is 360° (the entire circle), 360°: 2 = 180°
If an inscribed angle is based on the diameter of a given circle, its degree measure is 90°, since the diameter subtends an arc equal to 180°, 180°: 2 = 90°
If the central and inscribed angles in a circle rest on the same arc or chord, then the inscribed angle is equal to half the central one.

Where can problems on this topic be found? Their types and solutions

Since the circle and its properties are one of the most important sections of geometry, planimetry in particular, the inscribed and central angles in a circle are a topic that is studied widely and in detail in school course. Problems devoted to their properties are found mainly state exam(OGE) and the Unified State Exam (USE). As a rule, to solve these problems you need to find the angles on a circle in degrees.

Angles based on one arc

This type of problem is perhaps one of the easiest, since to solve it you need to know only two simple properties: if both angles are inscribed and rest on the same chord, they are equal; if one of them is central, then the corresponding inscribed angle is equal to half of it. However, when solving them you need to be extremely careful: sometimes it is difficult to notice this property, and students reach a dead end when solving such simple problems. Let's look at an example.

Task No. 1

Given a circle with center at point O. Angle AOB is 54°. Find the degree measure of angle ASV.

This task is solved in one action. The only thing you need to find the answer to it quickly is to notice that the arc on which both angles rest is common. Having seen this, you can apply an already familiar property. Angle ACB is equal to half of angle AOB. Means,

1) AOB = 54°: 2 = 27°.

Answer: 54°.

Angles subtended by different arcs of the same circle

Sometimes the problem conditions do not directly state the size of the arc on which the desired angle rests. In order to calculate it, you need to analyze the magnitude of these angles and compare them with known properties circles.

Problem 2

In a circle with center at point O, angle AOC is 120°, and angle AOB is 30°. Find the angle of YOU.

To begin with, it is worth saying that it is possible to solve this problem using the properties isosceles triangles, however, this will require running more mathematical operations. Therefore, here we will provide an analysis of the solution using the properties of central and inscribed angles in a circle.

So, angle AOS rests on arc AC and is central, which means arc AC is equal to angle AOS.

In the same way, angle AOB rests on arc AB.

Knowing this and the degree measure of the entire circle (360°), you can easily find the magnitude of the arc BC.

BC = 360° - AC - AB

BC = 360° - 120° - 30° = 210°

The vertex of angle CAB, point A, lies on the circle. This means that angle CAB is an inscribed angle and is equal to half of the arc NE.

Angle CAB = 210°: 2 = 110°

Answer: 110°

Problems based on the relationship of arcs

Some problems do not contain data on angle values at all, so you need to look for them based only on famous theorems and properties of a circle.

Problem 1

Find the angle inscribed in the circle that subtends the chord, equal to the radius given circle.

If you mentally draw lines connecting the ends of the segment to the center of the circle, you will get a triangle. Having examined it, you can see that these lines are the radii of the circle, which means that all sides of the triangle are equal. It is known that all angles equilateral triangle equal to 60°. This means that the arc AB containing the vertex of the triangle is equal to 60°. From here we find the arc AB on which the desired angle rests.

AB = 360° - 60° = 300°

Angle ABC = 300°: 2 = 150°

Answer: 150°

Problem 2

In a circle with a center at point O, the arcs are in a ratio of 3:7. Find the smallest inscribed angle.

To solve, let’s designate one part as X, then one arc is equal to 3X, and the second, respectively, is 7X. Knowing that the degree measure of a circle is 360°, let's create an equation.

3X + 7X = 360°

According to the condition, you need to find a smaller angle. Obviously, if the magnitude of the angle is directly proportional to the arc on which it rests, then the desired (smaller) angle corresponds to an arc equal to 3X.

This means that the smaller angle is (36° * 3) : 2 = 108°: 2 = 54°

Answer: 54°

In a circle with center at point O, angle AOB is 60°, and the length of the smaller arc is 50. Calculate the length of the larger arc.

In order to calculate the length of a larger arc, you need to create a proportion - how the smaller arc relates to the larger one. To do this, we calculate the magnitude of both arcs in degrees. The smaller arc is equal to the angle that rests on it. Its degree measure will be 60°. The major arc is equal to the difference between the degree measure of the circle (it is equal to 360° regardless of other data) and the minor arc.

The major arc is 360° - 60° = 300°.

Since 300°: 60° = 5, the larger arc is 5 times larger than the smaller one.

Large arc = 50 * 5 = 250

So, of course, there are other approaches to solving similar tasks, but all of them are in one way or another based on the properties of central and inscribed angles, triangles and circles. In order to successfully solve them, you need to carefully study the drawing and compare it with the data of the problem, and also be able to apply your theoretical knowledge in practice.

CIRCLE AND CIRCLE. CYLINDER.

§ 76. INSCRIBED AND SOME OTHER ANGLES.

1. Inscribed angle.

An angle whose vertex is on a circle and whose sides are chords is called an inscribed angle.

Angle ABC is an inscribed angle. It rests on the arc AC, enclosed between its sides (Fig. 330).

Theorem. An inscribed angle is measured by the half of the arc on which it subtends.

This should be understood this way: an inscribed angle contains as many angular degrees, minutes and seconds as there are arc degrees, minutes and seconds contained in the half of the arc on which it rests.

When proving this theorem, three cases must be considered.

First case. The center of the circle lies on the side of the inscribed angle (Fig. 331).

Let / ABC is an inscribed angle and the center of the circle O lies on side BC. It is required to prove that it is measured by half the arc AC.

Let's connect point A to the center of the circle. We get an isosceles /\ AOB, in which
AO = OB, as the radii of the same circle. Hence, / A = / IN. / AOC is external to triangle AOB, therefore / AOC = / A+ / B (§ 39, paragraph 2), and since angles A and B are equal, then / B is 1/2 / AOC.

But / AOC is measured by arc AC, therefore, / B is measured by half the arc AC.

For example, if the AC contains 60° 18", then / B contains 30°9".

Second case. The center of the circle lies between the sides of the inscribed angle (Fig. 332).

Let / ABD - inscribed angle. The center of circle O lies between its sides. It is required to prove that / ABD is measured by half the arc AD.

To prove this, let’s draw the diameter of the sun. Angle ABD is split into two angles: / 1 and / 2.

/ 1 is measured by half an arc AC, and / 2 is measured by half of the arc CD, therefore the entire / ABD is measured by 1/2 AC + 1/2 CD, i.e. half the arc AD.
For example, if AD contains 124°, then / B contains 62°.

Third case. The center of the circle lies outside the inscribed angle (Fig. 333).

Let / MAD - inscribed angle. The center of circle O is outside the corner. It is required to prove that / MAD is measured by half the arc MD.

To prove this, let’s draw the diameter AB. / MAD = / MAV- / DAB. But / MAV is measured at 1/2 MV, and / DAB is measured as 1/2 DB. Hence, / MAD is measured
1/2 (MB - DB), i.e. 1/2 MD.
For example, if MD contains 48° 38"16", then / MAD contains 24° 19" 8".

Consequences. 1. All inscribed angles subtending the same arc are equal to each other, since they are measured by half of the same arc (Figure 334, a).

2. An inscribed angle subtended by a diameter is a right angle, since it subtends half a circle. Half a circle contains 180 arc degrees, which means that the angle based on the diameter contains 90 arc degrees (Fig. 334, b).

2. The angle formed by a tangent and a chord.

Theorem. The angle formed by a tangent and a chord is measured by half the arc enclosed between its sides.

Let / CAB is composed of chord CA and tangent AB (Fig. 335). It is required to prove that it is measured by half the SA. Let's draw a straight line CD through point C || AB. Inscribed / ACD is measured by half of the arc AD, but AD = CA, since they are contained between the tangent and the chord parallel to it. Hence, / DCA is measured by half the arc of CA. Since this / CAB = / DCA, then it is measured by half the arc CA.

Exercises.

1. In drawing 336, find the tangents to the circle of the blocks.

2. According to drawing 337, prove that angle ADC is measured by half the sum of arcs AC and BC.

3. Using drawing 337, b, prove that angle AMB is measured by the half-difference of arcs AB and CE.