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 be 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 an arc AC.
Let's connect point A to the center of the circle. We obtain an isosceles \(\Delta\)AOB, in which AO = OB, as the radii of the same circle. Therefore, ∠A = ∠B.
∠AOC is external to triangle AOB, so ∠AOC = ∠A + ∠B, 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 of arc AC.
For example, if \(\breve(AC)\) 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 be an inscribed angle. The center of circle O lies between its sides. We need to prove that ∠ABD is measured by half the arc AD.
To prove this, let us draw the diameter BC. Angle ABD is split into two angles: ∠1 and ∠2.
∠1 is measured by half an arc AC, and ∠2 is measured by half an arc CD, therefore, the entire ∠ABD is measured by 1 / 2 \(\breve(AC)\) + 1 / 2 \(\breve(CD)\), i.e. . half arc AD.
For example, if \(\breve(AD)\) contains 124°, then ∠B contains 62°.
Third case. The center of the circle lies outside the inscribed angle (Fig. 333).
Let ∠MAD be an inscribed angle. The center of circle O is outside the corner. We need to prove that ∠MAD is measured by half the arc MD.
To prove this, let's draw the diameter AB. ∠MAD = ∠MAB - ∠DAB. But ∠MAB measures 1 / 2 \(\breve(MB)\), and ∠DAB measures 1 / 2 \(\breve(DB)\).
Therefore, ∠MAD measures 1 / 2 (\(\breve(MB) - \breve(DB))\), i.e. 1 / 2 \(\breve(MD)\).
For example, if \(\breve(MD)\) contains 48° 38", 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
(Fig. 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).
Intermediate level
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 the radius is equal to half the 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, a larger arc corresponds to a 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 the 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 more accurately calculate this mysterious number, to better 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! We recall that an external angle is equal to the sum of two internal angles 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’s a painfully 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 matter, 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 a 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 you 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.
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 subtended by 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 rests on the arc AKB and is equal to half the angular value 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°.
Inscribed angle, theory of the problem. Friends! In this article we will talk about tasks for which you need to know the properties of an inscribed angle. This is a whole group of tasks, they are included in the Unified State Exam. Most of them can be solved very simply, in one action.
There are more difficult problems, but they won’t present much difficulty for you; you need to know the properties of an inscribed angle. Gradually we will analyze all the prototypes of tasks, I invite you to the blog!
Now the necessary theory. Let us remember what a central and inscribed angle, a chord, an arc are, on which these angles rest:
The central angle in a circle is a plane angle withapex at its center.
The part of a circle located inside a plane anglecalled an arc of a circle.
The degree measure of an arc of a circle is called degree measure the corresponding central angle.
An angle is said to be inscribed in a circle if the vertex of the angle lieson a circle, and the sides of the angle intersect this circle.
A segment connecting two points on a circle is calledchord. The largest chord passes through the center of the circle and is calleddiameter.
To solve problems involving 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 subtending the same arc are equal.
3. All inscribed angles based on the same chord and whose vertices 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: the opposite angles of a quadrilateral inscribed in a circle add up to 180 degrees.
5. All inscribed angles subtended by a diameter are right angles.
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 unfolded angle is nothing more than a diameter), which means, according to the first property, the inscribed angle C is equal to half of it, that is, 90 degrees.
Knowing 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 the problems of this type orally. Two conclusions that can be drawn:
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 middle of its hypotenuse.
Many prototypes of stereometric problems are also solved by using this property and these consequences. 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 other conclusions and consequences yourself; you don’t need to teach them.
As a rule, half of the problems on inscribed angles are given with a sketch, but without symbols. To understand the reasoning process when solving problems (below in the article), notations for vertices (angles) are introduced. You don't have to do this on the Unified State Examination.Let's consider the tasks:
What is the value of an acute inscribed angle subtended by a chord equal to the radius of the circle? Give your answer in degrees.
Let's construct a central angle for a given inscribed angle and designate the vertices:
According to the property of an angle inscribed in a circle:
Angle AOB is equal to 60 0, since the triangle AOB is equilateral, and in 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 ACB is equal to 30 0.
Answer: 30
Find the chord supported by an angle of 30 0 inscribed in a circle of radius 3.
This is essentially the inverse problem (of the previous one). Let's construct the central angle.
It is twice as large as the inscribed one, that is, angle AOB is equal to 60 0. From this we can conclude that 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 magnitude of the obtuse inscribed angle subtended by the chord equal to the root of two. Give your answer in degrees.
Let's construct the central angle:
Knowing the radius and chord, we can find the central angle ASV. This can be done using the cosine theorem. Knowing the central angle, we can easily find the inscribed angle ACB.
Cosine theorem: square any side of the triangle 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 .
Angle ACB, by the property of an inscribed angle, is equal to half of it, that is, 135 degrees.
Answer: 135
Find the chord subtended by an angle of 120 degrees inscribed in a circle of radius root of three.
Connect points A and B to the center of the circle. Let's denote it as O:
We know the radius and inscribed angle ASV. 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.
According to the property of the 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 angle AOB in triangle AOB is equal to 360 0 – 240 0 = 120 0.
According to the cosine theorem:
Answer:3
Find the inscribed angle subtended by an arc that is 20% of the circle. Give your answer in degrees.
According to the property of the inscribed angle, it is half the size of the central angle based on the same arc, in in this case We are talking about arc AB.
It is said that 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 an angle of 360 degrees. Means,
Thus, the inscribed angle ACB is 36 degrees.
Answer: 36
Arc of a circle A.C., not containing a point B, is 200 degrees. And the arc of a circle BC, not containing a point A, is 80 degrees. Find the inscribed angle ACB. Give your answer in degrees.
For clarity, let us denote the arcs whose angular measures are given. Arc corresponding to 200 degrees – blue, the arc corresponding to 80 degrees is red, the remaining part of the circle is yellow.
Thus, the degree measure of the arc AB (yellow), and therefore the central angle AOB is: 360 0 – 200 0 – 80 0 = 80 0 .
The inscribed angle ACB is half the size of the central angle AOB, that is, equal to 40 degrees.
Answer: 40
What is the inscribed angle subtended by the diameter of the circle? Give your answer in degrees.
Today we will look at another type of problems 6 - this time with a circle. Many students do not like them and find them difficult. And completely in vain, since such problems are solved elementary, if you know some theorems. Or they don’t dare at all if you don’t know them.
Before talking about the main properties, let me remind you of the definition:
An inscribed angle is one whose vertex lies on the circle itself, and whose sides cut out a chord on this circle.
A central angle is any angle with its vertex at the center of the circle. Its sides also intersect this circle and carve a chord on it.
So, the concepts of inscribed and central angles are inextricably linked with the circle and the chords inside it. And now the main statement:
Theorem. The central angle is always twice the inscribed angle, based on the same arc.
Despite the simplicity of the statement, there is a whole class of problems 6 that can be solved using it - and nothing else.
Task. Find an acute inscribed angle subtended by a chord equal to the radius of the circle.
Let AB be the chord under consideration, O the center of the circle. Additional construction: OA and OB are the radii of the circle. We get:
Consider triangle ABO. In it AB = OA = OB - all sides are equal to the radius of the circle. Therefore, triangle ABO is equilateral, and all angles in it are 60°.
Let M be the vertex of the inscribed angle. Since angles O and M rest on the same arc AB, the inscribed angle M is 2 times smaller than the central angle O. We have:
M = O: 2 = 60: 2 = 30
Task. The central angle is 36° greater than the inscribed angle subtended by the same arc of a circle. Find the inscribed angle.
Let us introduce the following notation:
- AB is the chord of the circle;
- Point O is the center of the circle, so angle AOB is the central angle;
- Point C is the vertex of the inscribed angle ACB.
Since we are looking for the inscribed angle ACB, let's denote it ACB = x. Then the central angle AOB is x + 36. On the other hand, the central angle is 2 times the inscribed angle. We have:
AOB = 2 · ACB ;
x + 36 = 2 x ;
x = 36.
So we found the inscribed angle AOB - it is equal to 36°.
A circle is an angle of 360°
Having read the subtitle, knowledgeable readers will probably now say: “Ugh!” Indeed, comparing a circle with an angle is not entirely correct. To understand what we're talking about, take a look at the classic trigonometric circle:
What is this picture for? And besides, a full rotation is an angle of 360 degrees. And if you divide it, say, into 20 equal parts, then the size of each of them will be 360: 20 = 18 degrees. This is exactly what is required to solve problem B8.
Points A, B and C lie on the circle and divide it into three arcs, the degree measures of which are in the ratio 1: 3: 5. Find the greater angle of triangle ABC.
First, let's find the degree measure of each arc. Let the smaller one be x. In the figure this arc is designated AB. Then the remaining arcs - BC and AC - can be expressed in terms of AB: arc BC = 3x; AC = 5x. In total, these arcs give 360 degrees:
AB + BC + AC = 360;
x + 3x + 5x = 360;
9x = 360;
x = 40.
Now consider a large arc AC that does not contain point B. This arc, like the corresponding central angle AOC, is 5x = 5 40 = 200 degrees.
Angle ABC is the largest of all angles in a triangle. It is an inscribed angle subtended by the same arc as the central angle AOC. This means that angle ABC is 2 times smaller than AOC. We have:
ABC = AOC: 2 = 200: 2 = 100
This will be the degree measure of the larger angle in triangle ABC.
Circle circumscribed around a right triangle
Many people forget this theorem. But in vain, because some B8 problems cannot be solved at all without it. More precisely, they are solved, but with such a volume of calculations that you would rather fall asleep than reach the answer.
Theorem. The center of a circle circumscribed around a right triangle lies at the midpoint of the hypotenuse.
What follows from this theorem?
- The midpoint of the hypotenuse is equidistant from all the vertices of the triangle. This is a direct consequence of the theorem;
- The median drawn to the hypotenuse divides the original triangle into two isosceles triangles. This is exactly what is required to solve problem B8.
In triangle ABC we draw the median CD. Angle C is 90° and angle B is 60°. Find angle ACD.
Since angle C is 90°, triangle ABC is a right triangle. It turns out that CD is the median drawn to the hypotenuse. This means that triangles ADC and BDC are isosceles.
In particular, consider triangle ADC. In it AD = CD. But in an isosceles triangle, the angles at the base are equal - see “Problem B8: Line segments and angles in triangles.” Therefore, the desired angle ACD = A.
So, it remains to find out what the angle A is equal to. To do this, let's turn again to the original triangle ABC. Let's denote the angle A = x. Since the sum of the angles in any triangle is 180°, we have:
A + B + BCA = 180;
x + 60 + 90 = 180;
x = 30.
Of course, the last problem can be solved differently. For example, it is easy to prove that triangle BCD is not just isosceles, but equilateral. So angle BCD is 60 degrees. Hence angle ACD is 90 − 60 = 30 degrees. As you can see, you can use different isosceles triangles, but the answer will always be the same.