Angle bisector. Complete Lessons - Knowledge Hypermarket

Today will be a very easy lesson. We will consider only one object - the bisector of an angle - and prove its most important property, which will be very useful to us in the future.

Just do not relax: sometimes students who want to get a high score on the same OGE or USE, in the first lesson, cannot even accurately formulate the definition of the bisector.

And instead of doing really interesting tasks, we waste time on such simple things. Therefore, read, see - and take it into service. :)

For starters, a little strange question: what is an angle? That's right: an angle is just two rays coming out of the same point. For example:

Examples of angles: sharp, obtuse and straight

As you can see from the picture, the corners can be sharp, obtuse, straight - it doesn't matter now. Often, for convenience, an additional point is marked on each ray and they say that we have an angle $ AOB $ in front of us (written as $ \ angle AOB $).

The captain of the obviousness seems to be hinting that in addition to the $ OA $ and $ OB $ rays, you can always draw a bunch of rays from the $ O $ point. But among them there will be one special - it is he who is called the bisector.

Definition. The bisector of an angle is a ray that exits from the top of that angle and bisects the angle.

For the above angles, the bisectors will look like this:

Examples of bisectors for acute, obtuse and right angles

Since in real drawings it is far from always obvious that a certain ray (in our case it is the $ OM $ ray) splits the initial angle into two equal angles, in geometry it is customary to mark equal angles with the same number of arcs (in our drawing, this is 1 arc for an acute angle, two for blunt, three for direct).

Okay, we've figured out the definition. Now you need to understand what properties the bisector has.

The main property of the bisector of an angle

In fact, a bisector has a bunch of properties. And we will definitely look at them in the next lesson. But there is one trick that you need to understand right now:

Theorem. The bisector of an angle is the locus of points equidistant from the sides of a given angle.

Translated from mathematical into Russian, this means two facts at once:

Any point lying on the bisector of a certain angle is at the same distance from the sides of this angle.
And vice versa: if a point lies at the same distance from the sides of a given angle, then it is guaranteed to lie on the bisector of this angle.

Before proving these statements, let's clarify one point: what, in fact, is called the distance from a point to a side of an angle? Here, the good old-fashioned definition of the distance from a point to a line will help us:

Definition. The distance from a point to a line is the length of a perpendicular drawn from a given point to that line.

For example, consider the line $ l $ and a point $ A $ that does not lie on this line. Draw a perpendicular $ AH $, where $ H \ in l $. Then the length of this perpendicular will be the distance from the point $ A $ to the straight line $ l $.

Graphical representation of distance from point to line

Since an angle is just two beams, and each beam is a piece of a straight line, it is easy to determine the distance from a point to the sides of the angle. They are just two perpendiculars:

Determine the distance from the point to the sides of the corner

That's all! Now we know what distance is and what a bisector is. Therefore, the main property can be proved.

As promised, let's split the proof into two parts:

1. Distances from a point on the bisector to the sides of the angle are the same

Consider an arbitrary angle with vertex $ O $ and bisector $ OM $:

Let us prove that this very point $ M $ is at the same distance from the sides of the corner.

Proof. Draw perpendiculars from the point $ M $ to the sides of the corner. Let's call them $ M ((H) _ (1)) $ and $ M ((H) _ (2)) $:
Draw perpendiculars to the sides of the corner
We got two right triangles: $ \ vartriangle OM ((H) _ (1)) $ and $ \ vartriangle OM ((H) _ (2)) $. They have a common hypotenuse $ OM $ and equal angles:

$ \ angle MO ((H) _ (1)) = \ angle MO ((H) _ (2)) $ by condition (since $ OM $ is a bisector);

$ \ angle M ((H) _ (1)) O = \ angle M ((H) _ (2)) O = 90 () ^ \ circ $ by construction;

$ \ angle OM ((H) _ (1)) = \ angle OM ((H) _ (2)) = 90 () ^ \ circ - \ angle MO ((H) _ (1)) $, since the sum The acute angles of a right triangle are always 90 degrees.

Consequently, triangles are equal in side and two adjacent angles (see signs of equality of triangles). Therefore, in particular, $ M ((H) _ (2)) = M ((H) _ (1)) $, i.e. the distances from the point $ O $ to the sides of the corner are indeed equal. Q.E.D.:)

2. If the distances are equal, then the point lies on the bisector

Now the situation is reversed. Let an angle $ O $ and a point $ M $ equidistant from the sides of this angle be given:

Let us prove that the ray $ OM $ is a bisector, that is, $ \ angle MO ((H) _ (1)) = \ angle MO ((H) _ (2)) $.

Proof. To begin with, let's draw this very ray $ OM $, otherwise there will be nothing to prove:
Spent the $ OM $ ray inside the corner
Again we got two right triangles: $ \ vartriangle OM ((H) _ (1)) $ and $ \ vartriangle OM ((H) _ (2)) $. They are obviously equal because:

Hypotenuse $ OM $ - total;

The legs $ M ((H) _ (1)) = M ((H) _ (2)) $ by condition (after all, the point $ M $ is equidistant from the sides of the corner);

The remaining legs are also equal, because by the Pythagorean theorem $ OH_ (1) ^ (2) = OH_ (2) ^ (2) = O ((M) ^ (2)) - MH_ (1) ^ (2) $.

Therefore, the triangles $ \ vartriangle OM ((H) _ (1)) $ and $ \ vartriangle OM ((H) _ (2)) $ are on three sides. In particular, their angles are equal: $ \ angle MO ((H) _ (1)) = \ angle MO ((H) _ (2)) $. And this just means that $ OM $ is a bisector.

In conclusion of the proof, we mark the resulting equal angles with red arcs:
The bisector split the $ \ angle ((H) _ (1)) O ((H) _ (2)) $ into two equal

As you can see, nothing complicated. We have proved that the bisector of an angle is the locus of points equidistant to the sides of this angle. :)

Now that we have more or less decided on the terminology, it's time to move to a new level. In the next lesson, we will analyze the more complex properties of the bisector and learn how to use them to solve real problems.

The bisector of a triangle is a segment that divides the angle of the triangle into two equal angles. For example, if the angle of the triangle is 120 0, then drawing the bisector, we will build two angles 60 0 each.

And since there are three angles in a triangle, three bisectors can be drawn. They all have one cut-off point. This point is the center of the circle inscribed in the triangle. In another way, this intersection point is called the incenter of the triangle.

When two bisectors of the inner and outer angles intersect, an angle of 90 0 is obtained. The outer corner in a triangle is the angle adjacent to the inner corner of the triangle.

Rice. 1. Triangle with 3 bisectors

The bisector divides the opposite side into two line segments that are connected to the sides:

$$ (CL \ over (LB)) = (AC \ over (AB)) $$

The points of the bisector are equidistant from the sides of the corner, which means that they are at the same distance from the sides of the corner. That is, if from any point of the bisector we lower the perpendiculars to each side of the angle of the triangle, then these perpendiculars will be equal.

If you draw the median, bisector and height from one vertex, then the median will be the longest segment, and the height will be the shortest.

Some properties of the bisector

In certain types of triangles, the bisector has special properties. This primarily applies to the isosceles triangle. This figure has two identical sides, and the third is called the base.

If from the vertex of the angle of an isosceles triangle to draw a bisector to the base, then it will have the properties of both the height and the median. Accordingly, the length of the bisector coincides with the length of the median and height.

Definitions:

Height- the perpendicular dropped from the apex of the triangle to the opposite side ..
Median- a segment that connects the top of the triangle and the middle of the opposite side.

Rice. 2. Bisector in an isosceles triangle

This also applies to an equilateral triangle, that is, a triangle in which all three sides are equal.

Example task

In a triangle ABC: BR is the bisector, where AB = 6 cm, BC = 4 cm, and RC = 2 cm. Subtract the length of the third side.

Rice. 3. Bisector in a triangle

Solution:

The bisector divides the side of the triangle in a specific proportion. Let's use this proportion and express AR. Then we will find the length of the third side as the sum of the segments into which this side was divided by the bisector.

$ (AB \ over (BC)) = (AR \ over (RC)) $
$ RC = (6 \ over (4)) * 2 = 3 cm $

Then the entire segment AC = RC + AR

AC = 3 + 2 = 5 cm.

In an isosceles triangle, the bisector drawn to the base divides the triangle into two equal right-angled triangles.

What have we learned?

After studying the topic of the bisector, we learned that it divides an angle into two equal angles. And if it is drawn in an isosceles or equilateral triangle to the base, then it will have the properties of both the median and the height at the same time.

Test by topic

Article rating

Average rating: 4.2. Total ratings received: 157.

The bisector of a triangle is a common geometric concept that does not cause any particular difficulties in studying. Having knowledge of its properties, many problems can be solved without much difficulty. What is a bisector? We will try to acquaint the reader with all the secrets of this mathematical line.

In contact with

The essence of the concept

The name of the concept came from the use of words in Latin, the meaning of which is "bi" - two, "sectio" - cut. They specifically point to the geometric meaning of the concept - breaking up the space between the rays into two equal parts.

The bisector of a triangle is a segment that originates from the top of the figure, and the other end is located on the side opposite it, while dividing the space into two equal parts.

For quick associative memorization of mathematical concepts by students, many teachers use different terminology, which is displayed in verses or associations. Of course, this definition is recommended for older children.

How is this straight line designated? Here we rely on the rules for denoting segments or rays. If we are talking about the designation of the bisector of the angle of a triangular figure, then it is usually written as a segment, the ends of which are vertex and point of intersection with the side opposite to the vertex... Moreover, the beginning of the designation is written precisely from the top.

Attention! How many bisectors does a triangle have? The answer is obvious: there are as many as there are three peaks.

Properties

In addition to the definition, in the school textbook you can find not so many properties of this geometric concept. The first property of the bisector of a triangle, which is introduced to schoolchildren, is the center of the inscribed, and the second, directly related to it, is the proportionality of the segments. The bottom line is as follows:

Whatever the dividing line, there are points on it that are at the same distance from the sides that make up the space between the beams.
In order to inscribe a circle in a triangular figure, it is necessary to determine the point at which these segments will intersect. This is the center point of the circle.
The parts of the side of a triangular geometric figure, into which the dividing line splits it, are proportional to the angled sides.

We will try to bring the rest of the features into the system and present additional facts that will help to better understand the merits of this geometric concept.

Length

One of the types of problems that cause difficulties for schoolchildren is finding the length of the bisector of the angle of a triangle. The first option, which contains its length, contains the following data:

the amount of space between the rays, from the top of which this segment comes out;
the lengths of the sides that form this angle.

To solve the problem the formula is used, the meaning of which is to find the ratio of the doubled product of the values of the sides making up the angle by the cosine of its half to the sum of the sides.

Let's consider a specific example. Suppose a figure ABC is given, in which a segment is drawn from angle A and intersects the side BC at point K. The value of A is denoted by Y. Based on this, AK = (2 * AB * AC * cos (Y / 2)) / (AB + AC).

The second version of the problem, in which the length of the bisector of a triangle is determined, contains the following data:

the meanings of all sides of the figure are known.

When solving a problem of this type, initially determine the semi-perimeter... To do this, add the values of all sides and divide in half: p = (AB + BC + AC) / 2. Next, we apply the computational formula, which was used to determine the length of this segment in the previous problem. It is only necessary to make some changes to the essence of the formula in accordance with the new parameters. So, it is necessary to find the ratio of the doubled root of the second degree from the product of the lengths of the sides that adjoin the vertex by the half-perimeter and the difference between the half-perimeter and the length of the opposite side to the sum of the sides that make up the angle. That is, AK = (2٦AB * AC * p * (p-BC)) / (AB + AC).

Attention! To make it easier to master the material, you can refer to the comic tales available on the Internet that tell about the "adventures" of this straight line.

Special cases

The bisector of a right-angled triangle has all the general properties. But a special case should be noted, which is inherent only to it: when intersecting segments, the bases of which are the tops of acute right-angled triangles, 45 degrees are obtained between the rays.

The bisector of an isosceles triangle also has its own characteristics:

If the base of this segment is the top opposite to the base, then it is both height and median.
If the segments are drawn from the vertices of the corners at the base, then their lengths are equal to each other.

Geometry lesson, we study the properties of the bisector

Properties of the bisector of a triangle

what is the angle bisector?

Besectrix is a rat that walks in corners and bisects the corner.
Bisector properties

a2a1 = cb
la = c + bcb (b + c + a) (b + ca)
la = c + b2bc cos2
la = hacos2
la = bca1a2
Where:
so somehow))
The flat angle of the unfolded angle divides it into 2 right angles
this rat splits
The bisector (from Latin bi - double, and sectio cutting) of the angle is a ray with the beginning at the apex of the angle, dividing the angle into two equal parts.
The bisector (from Latin bi - double, and sectio cutting) of the angle is a ray with the beginning at the apex of the angle, dividing the angle into two equal parts.
The bisector is a rat that runs in corners and divides the corner by sex.
beam dividing angle into 2 equal angles
The bisector is a rat that runs around corners and bisects the corner!
😉
The bisector (from Latin bi - double, and sectio cutting) of the angle is a ray with the beginning at the apex of the angle, dividing the angle into two equal parts.
The bisector of an angle (together with its continuation) is the locus of points equidistant from the sides of the angle (or their extensions).
Definition. The bisector of an angle of a triangle is a segment of the bisector of this angle that connects this vertex to a point on the opposite side.
Any of the three bisectors of the interior angles of a triangle is called the bisector of the triangle.
The bisector of an angle of a triangle can denote one of two things: the ray is the bisector of this angle or a segment of the bisector of this angle before it intersects with the side of the triangle.
Bisector properties
The angle bisector of a triangle divides the opposite side in a ratio equal to the ratio of the two adjacent sides.
The bisectors of the inner corners of the triangle intersect at one point. This point is called the center of the inscribed circle.
The bisectors of the inner and outer corners are perpendicular.
If the bisector of the outer corner of the triangle intersects the continuation of the opposite side, then ADBD = ACBC.
The bisectors of one inner and two outer corners of a triangle intersect at one point. This point is the center of one of the three excircles of this triangle.
The bases of the bisectors of two inner and one outer corners of a triangle lie on the same straight line if the bisector of the outer corner is not parallel to the opposite side of the triangle.
If the bisectors of the outer corners of the triangle are not parallel to the opposite sides, then their bases lie on one straight line.
a2a1 = cb
la = c + bcb (b + c + a) (b + c # 8722; a)
la = c + b2bc cos2
la = hacos2 # 8722;
la = bc # 8722; a1a2
Where:
la is the bisector to side a,
a, b, from the side of the triangle against the vertices A, B, C, respectively,
al, a 2 segments into which the bisector lc divides side c,
the interior angles of the triangle at the vertices a, b, c, respectively,
ha is the height of the triangle dropped to side a.
the bisector is a line that divides the angle by palam
The bisector (from Latin bi - double, and sectio cutting) of the angle is a ray with the beginning at the apex of the angle, dividing the angle into two equal parts.
The bisector of an angle (together with its continuation) is the locus of points equidistant from the sides of the angle (or their extensions).
The bisector is a rat that walks in corners, halves the angle.
the bisector, such a rat, runs around the corners and divides the angle into hits)
Divides the corner in half
the line that divides it (the corner) in half.
The bisector is a rat that runs around the corners and divides them in half.

A bisector is a line that halves an angle.

Did you meet the bisector in the problem? Try to apply one (and sometimes several) of the following amazing properties.

1. Bisector in an isosceles triangle.

Aren't you afraid of the word "theorem"? If you are afraid, then - in vain. Mathematicians are accustomed to calling any statement that can somehow be deduced from other, simpler statements by a theorem.

So, attention, theorem!

Let's prove this theorem, that is, we will understand why this is so? Look at the isosceles.

Let's take a close look at them. And then we will see that

- general.

And this means (rather, remember the first sign of equality of triangles!) That.

So what? Do you want to say so? And the fact that we have not yet looked at the third sides and the remaining corners of these triangles.

Now let's see. Once, then absolutely exactly and even in addition,.

So it turned out that

divided the side in half, that is, it turned out to be the median
, which means that they are both on, because (take another look at the picture).

So it turned out to be a bisector and a height too!

Hooray! We have proved the theorem. But imagine, that's not all. It is also true converse theorem:

Proof? Are you wondering? Read the next level of theory!

And if not interesting, then remember firmly:

Why memorize this firmly? How can this help? But imagine that you have a task:

Given: .

Find: .

You immediately realize, bisector and, lo and behold, she divided the side in half! (by condition…). If you firmly remember that it happens only in an isosceles triangle, then you conclude what it means, you write the answer:. Great, isn't it? Of course, not all tasks will be so easy, but knowledge will definitely help!

And now the next property. Ready?

2. The bisector of an angle is the locus of points equidistant from the sides of the angle.

Scared? Actually, it's okay. Lazy mathematicians hid four in two lines. So, what does it mean, "Bisector - locus of points"? This means that they are executed immediately. twostatements:

If the point lies on the bisector, then the distances from it to the sides of the angle are equal.
If at some point the distances to the sides of the corner are equal, then this point necessarily lies on the bisector.

Do you see the difference between statements 1 and 2? If not, then remember the Hatter from Alice in Wonderland: "So you still have something good to say, as if" I see what I eat "and" I eat what I see "are one and the same!"

So, we need to prove statements 1 and 2, and then the statement: "the bisector is the locus of points equidistant from the sides of the corner" will be proved!

Why is 1 true?

Take any point on the bisector and name it.

Let us drop the perpendiculars from this point to the sides of the corner.

And now ... get ready to remember the signs of equality of right-angled triangles! If you have forgotten them, then take a look at the section.

So ... two right-angled triangles: and. They have:

General hypotenuse.
(because - bisector!)

This means - by angle and hypotenuse. Therefore, the corresponding legs of these triangles are equal! That is.

It was proved that the point is equally (or equal) distant from the sides of the corner. With point 1 sorted out. Now let's move on to point 2.

Why is 2 true?

And connect the dots and.

So, that is, lies on the bisector!

That's all!

How can all this be applied to solving problems? For example, in problems there is often such a phrase: “The circle touches the sides of the corner….”. Well, and you need to find something.

You quickly realize that

And you can use equality.

3. Three bisectors in a triangle intersect at one point

From the property of the bisector to be the locus of points equidistant from the sides of the angle, the following statement follows:

How exactly does it follow? But look: two bisectors will definitely intersect, right?

And the third bisector could go like this:

But in fact, everything is much better!

Let's consider the intersection point of two bisectors. Let's call it.

What have we used here both times? Yes paragraph 1, of course! If the point lies on the bisector, then it is equally distant from the sides of the corner.

So it turned out and.

But look carefully at these two equalities! After all, it follows from them that and, therefore,.

But now it will go into action point 2: if the distances to the sides of the angle are equal, then the point lies on the bisector ... what is the angle? Look at the picture again:

and are the distances to the sides of the angle, and they are equal, which means that the point lies on the bisector of the angle. The third bisector went through the same point! All three bisectors intersect at one point! And, as an additional gift -

Radius inscribed circles.

(To be sure, see another topic).

Well, now you will never forget:

The intersection point of the bisectors of a triangle is the center of the inscribed circle.

Moving on to the next property ... Wow, and the bisector has a lot of properties, right? And this is great, because the more properties, the more tools for solving problems about the bisector.

4. Bisector and parallelism, bisectors of adjacent angles

The fact that the bisector divides the angle in half, in some cases, leads to completely unexpected results. For example,

Case 1

Great, isn't it? Let's understand why this is so.

On the one hand, we are doing the bisector!

But, on the other hand, like criss-crossing corners (remember the topic).

And now it turns out that; throw out the middle:! - isosceles!

Case 2

Imagine a triangle (or look at the picture)

Let's continue the side for a point. Now we have two corners:

- inner corner
- the outer corner - it's outside, right?

So, now someone wanted to draw not one, but two bisectors at once: for and for. What will happen?

And it will turn out rectangular!

Surprisingly, this is exactly the case.

Understanding.

What do you think is the sum?

Of course, because they all together make up such an angle that it turns out to be a straight line.

And now remember that and are bisectors and see that inside the corner there is exactly half from the sum of all four angles: and - - that is, exactly. You can also write the equation:

So, unbelievable, but true:

The angle between the bisectors of the inner and outer corner of the triangle is.

Case 3

Do you see that everything is the same here as for the inner and outer corners?

Or think again why this is so?

Again, as for adjacent corners,

(as matched on parallel bases).

And again, make up exactly half from the sum

Output: If the problem contains bisectors related angles or bisectors the respective angles of a parallelogram or trapezoid, then in this problem certainly a right-angled triangle is involved, and maybe even a whole rectangle.

5. Bisector and opposite side

It turns out that the bisector of the angle of the triangle divides the opposite side not somehow, but in a special and very interesting way:

That is:

An amazing fact, isn't it?

Now we will prove this fact, but get ready: it will be a little more difficult than before.

Again - spacewalk - additional construction!

Let's draw a straight line.

What for? We’ll see now.

Continue the bisector to the intersection with the straight line.

Sound familiar? Yes, yes, yes, in the same way as in paragraph 4, case 1 - it turns out that (is the bisector)

Like lying crosswise

Means - this too.

Now let's look at the triangles and.

What can you say about them?

They are similar. Well, yes, they have the same angles as vertical. Hence, in two corners.

Now we have the right to write the relationship of the respective parties.

And now in short notation:

Ouch! Looks like something, right? Isn't that what we wanted to prove? Yes, that's it!

You see how great the "spacewalk" - the construction of an additional straight line - has proved itself - nothing would have happened without it! And so, we have proved that

Now you can safely use it! Let's analyze one more property of the bisectors of the angles of a triangle - don't be alarmed, now the hardest part is over - it will be easier.

We get that

This knowledge can be applied in those problems where two bisectors are involved and only the angle is given, and the required values are maintained through or, conversely, given, but you need to find something with the participation of the angle.

The basic knowledge of the bisector is over. By combining these facts, you will find the key to any bisector problem!

BISECTOR. SUMMARY AND BASIC FORMULAS

Theorem 1:

Theorem 2:

Theorem 3:

Theorem 4:

Theorem 5:

Theorem 6: