Geometry is one of the most complex and confusing sciences. In it, what seems obvious at first glance very rarely turns out to be correct. Bisectors, altitudes, medians, projections, tangents - a huge number of really difficult terms, which are very easy to get confused.
In fact, with the proper desire, you can understand a theory of any complexity. When it comes to bisectors, medians, and altitudes, you need to understand that they are not unique to triangles. At first glance this simple lines, but each of them has its own properties and functions, knowledge of which greatly simplifies the solution geometric problems. So, what is the bisector of a triangle?
Definition
The term "bisector" itself comes from the combination Latin words“two” and “cut”, “cut”, which already indirectly indicates its properties. Usually, when children are introduced to this ray, they are given a short phrase to remember: “The bisector is a rat that runs around the corners and divides the corner in half.” Naturally, such an explanation is not suitable for older schoolchildren; moreover, they are usually asked not about coal, but about geometric figure. So the bisector of a triangle is the ray that connects the vertex of the triangle to opposite side, while dividing the angle into two equal parts. The point on the opposite side at which the bisector comes for arbitrary triangle is selected randomly.
Basic functions and properties
This beam has few basic properties. First, because the bisector of a triangle bisects the angle, any point lying on it will be on equal distance from the sides forming the top. Secondly, in each triangle you can draw three bisectors, according to the number of available angles (hence, in the same quadrilateral there will already be four of them, and so on). The point at which all three rays intersect is the center of the circle inscribed in the triangle.
Properties become more complex
Let's complicate the theory a little. Another interesting property: the bisector of an angle of a triangle divides the opposite side into segments, the ratio of which is equal to the ratio of the sides forming the vertex. At first glance, this is complicated, but in fact everything is simple: in the proposed figure, RL: LQ = PR: PK. By the way, this property was called the “Bisector Theorem” and first appeared in the works of the ancient Greek mathematician Euclid. It was remembered in one of the Russian textbooks only in the first quarter of the seventeenth century.
It's a little more complicated. In a quadrilateral, the bisector cuts off an isosceles triangle. This figure shows all equal angles for the median AF.
And in quadrilaterals and trapezoids, the bisectors of one-sided angles are perpendicular to each other. In the drawing shown, angle APB is 90 degrees.
In an isosceles triangle
The bisector of an isosceles triangle is a much more useful ray. It is at the same time not only a divisor of an angle in half, but also a median and an altitude.
The median is a segment that comes from some corner and falls on the middle of the opposite side, thereby dividing it into equal parts. Height is a perpendicular descended from a vertex to the opposite side; it is with its help that any problem can be reduced to a simple and primitive Pythagorean theorem. In this situation, the bisector of the triangle is equal to the root of the difference between the square of the hypotenuse and the other leg. By the way, this property is most often encountered in geometric problems.
To consolidate: in this triangle, the bisector FB is the median (AB = BC) and the height (angles FBC and FBA are 90 degrees).
In outline
So what do you need to remember? The bisector of a triangle is the ray that bisects its vertex. At the intersection of three rays is the center of a circle inscribed in given triangle(the only disadvantage of this property is that it does not have practical value and serves only for the competent execution of the drawing). It also divides the opposite side into segments, the ratio of which is equal to the ratio of the sides between which this ray passed. In a quadrilateral, the properties become a little more complicated, but, admittedly, they practically never appear in problems school level, so they are usually not touched upon in the program.
The bisector of an isosceles triangle is the ultimate dream of any schoolchild. It is both a median (that is, it divides the opposite side in half) and an altitude (perpendicular to that side). Solving problems with such a bisector reduces to the Pythagorean theorem.
Knowledge basic functions bisector, as well as its basic properties, is necessary to solve geometric problems of both average and high level difficulties. In fact, this ray is found only in planimetry, so it cannot be said that memorizing information about it will allow you to cope with all types of tasks.
The bisector of a triangle is the segment that divides the angle of the triangle into two equal angles. For example, if the angle of a triangle is 120 0, then by drawing a bisector, we will construct two angles of 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 internal and external corner, the angle is 90 0. An exterior angle in a triangle is the angle adjacent to the interior angle of a triangle.
Rice. 1. A triangle containing 3 bisectors
The bisector divides the opposite side into two segments that are connected to the sides:
$$(CL\over(LB)) = (AC\over(AB))$$
The bisector points are equidistant from the sides of the angle, which means that they are at the same distance from the sides of the angle. That is, if from any point of the bisector we drop perpendiculars to each of the sides of the angle of the triangle, then these perpendiculars will be equal..
If you draw a 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 triangles, the bisector has special properties. This primarily applies to an isosceles triangle. This figure has two identical sides, and the third is called the base.
If you draw a bisector from the vertex of an angle of an isosceles triangle to the base, then it will have the properties of both height and median. Accordingly, the length of the bisector coincides with the length of the median and height.
Definitions:
- Height- a perpendicular drawn from the vertex of a triangle to the opposite side.
- Median– a segment that connects the vertex of a triangle and the middle of the opposite side.
Rice. 2. Bisector in an isosceles triangle
This also applies equilateral triangle, that is, a triangle in which all three sides are equal.
Example assignment
In triangle ABC: BR is a bisector, with 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 certain proportion. Let's use this proportion and express AR. Then we 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.
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Bisector of a triangle. Detailed theory with examples (2019)
Bisector of a triangle and its properties
Do you know what the midpoint of a segment is? Of course you do. What about the center of the circle? Same. What is the midpoint of an angle? You can say that this doesn't happen. But why can a segment be divided in half, but an angle cannot? It’s quite possible - just not a dot, but…. line.
Do you remember the joke: a bisector is a rat that runs around the corners and divides the corner in half. So, the real definition of a bisector is very similar to this joke:
Bisector of a triangle- this is the bisector segment of an angle of a triangle connecting the vertex of this angle with a point on the opposite side.
Once upon a time, ancient astronomers and mathematicians discovered a lot interesting properties bisectors. This knowledge has greatly simplified people's lives. It has become easier to build, count distances, even adjust the firing of cannons... Knowledge of these properties will help us solve some GIA and Unified State Examination tasks!
The first knowledge that will help with this is bisector of an isosceles triangle.
By the way, do you remember all these terms? Do you remember how they differ from each other? No? Not scary. Let's figure it out now.
So, base of an isosceles triangle- this is the side that is not equal to any other. Look at the picture, which side do you think it is? That's right - this is the side.
The median is a line drawn from the vertex of a triangle and dividing the opposite side (that's it again) in half.
Notice we don't say, "Median of an isosceles triangle." Do you know why? Because a median drawn from a vertex of a triangle bisects the opposite side in ANY triangle.
Well, the height is a line drawn from the top and perpendicular to the base. You noticed? We are again talking about any triangle, not just an isosceles one. The height in ANY triangle is always perpendicular to the base.
So, have you figured it out? Almost. To understand even better and forever remember what a bisector, median and height are, you need to compare them with each other and understand how they are similar and how they differ from each other. At the same time, in order to remember better, it is better to describe everything “ human language" Then you will easily operate in the language of mathematics, but at first you do not understand this language and you need to comprehend everything in your own language.
So, how are they similar? The bisector, the median and the altitude - they all “come out” from the vertex of the triangle and rest on the opposite side and “do something” either with the angle from which they come out, or with the opposite side. I think it's simple, no?
How are they different?
- The bisector divides the angle from which it emerges in half.
- The median divides the opposite side in half.
- The height is always perpendicular to the opposite side.
That's it. It's easy to understand. And once you understand, you can remember.
Now next question. Why in the case of isosceles triangle Is the bisector both the median and the height?
You can just look at the figure and make sure that the median divides into two absolutely equal triangle. That's all! But mathematicians do not like to believe their eyes. They need to prove everything. Scary word? Nothing like that - it's simple! Look: both have equal sides and, they generally have a common side and. (- bisector!) And so it turns out that two triangles have two equal sides and the angle between them. We recall the first sign of equality of triangles (if you don’t remember, look in the topic) and conclude that, and therefore = and.
This is already good - it means it turned out to be the median.
But what is it?
Let's look at the picture - . And we got it. So, too! Finally, hurray! And.
Did you find this proof a bit difficult? Look at the picture - two identical triangles speak for themselves.
In any case, remember firmly:
Now it’s more difficult: we’ll count angle between bisectors in any triangle! Don't be afraid, it's not that tricky. Look at the picture:
Let's count it. Do you remember that the sum of the angles of a triangle is?
Let's apply this amazing fact.
On the one hand, from:
That is.
Now let's look at:
But bisectors, bisectors!
Let's remember about:
Now through the letters
\angle AOC=90()^\circ +\frac(\angle B)(2)
Isn't it surprising? It turned out that the angle between the bisectors of two angles depends only on the third angle!
Well, we looked at two bisectors. What if there are three of them??!! Will they all intersect at one point?
Or will it be like this?
How do you think? So mathematicians thought and thought and proved:
Isn't that great?
Do you want to know why this happens?
So...two right triangles: and. They have:
- General hypotenuse.
- (because it is a bisector!)
This means - by angle and hypotenuse. Therefore, the corresponding legs of these triangles are equal! That is.
We proved that the point is equally (or equally) distant from the sides of the angle. Point 1 is dealt with. Now let's move on to point 2.
Why is 2 true?
And let's connect the dots and.
This means that it lies on the bisector!
That's all!
How can all this be applied when solving problems? For example, in problems there is often the following phrase: “A circle touches the sides of an angle...”. Well, you need to find something.
Then you quickly realize that
And you can use equality.
3. Three bisectors in a triangle intersect at one point
From the property of a bisector to be locus points equidistant from the sides of the angle, the following statement follows:
How exactly does it come out? But look: two bisectors will definitely intersect, right?
And the third bisector could go like this:
But in reality, everything is much better!
Let's look at the intersection point of two bisectors. Let's call it .
What did we use here both times? Yes paragraph 1, of course! If a point lies on a bisector, then it is equally distant from the sides of the angle.
And so it happened.
But look carefully at these two equalities! After all, it follows from them that and, therefore, .
And now it will come into play point 2: if the distances to the sides of the angle are equal, then the point lies on the bisector...what angle? Look at the picture again:
and are the distances to the sides of the angle, and they are equal, which means the point lies on the bisector of the angle. The third bisector passed through the same point! All three bisectors intersect at one point! And as an additional gift -
Radii inscribed circles.
(To be sure, look at another topic).
Well, now you'll never forget:
The point of intersection of the bisectors of a triangle is the center of the circle inscribed in it.
Let's move on to the next property... Wow, the bisector has many properties, right? And that's great, because more properties, the more tools for solving bisector problems.
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, right? Let's understand why this is so.
On the one hand, we draw a bisector!
But, on the other hand, there are angles that lie crosswise (remember the theme).
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 beyond the point. Now we have two angles:
- - internal corner
- - the outer corner is outside, right?
So, now someone wanted to draw not one, but two bisectors at once: both for and for. What will happen?
Will it work out? rectangular!
Surprisingly, this is exactly the case.
Let's figure it out.
What do you think the amount is?
Of course, - after all, they all together make such an angle that it turns out to be a straight line.
Now remember that and are bisectors and see that inside the angle there is exactly half from the sum of all four angles: and - - that is, exactly. You can also write it as an equation:
So, incredible but true:
The angle between the bisectors of the internal and external angles of a triangle is equal.
Case 3
Do you see that everything is the same here as for the internal and external corners?
Or let's think again why this happens?
Again, as for adjacent corners,
(as corresponding with parallel bases).
And again, they make up exactly half from the sum
Conclusion: If the problem contains bisectors adjacent angles or bisectors relevant angles of a parallelogram or trapezoid, then in this problem certainly participates right triangle, and maybe even a whole rectangle.
5. Bisector and opposite side
It turns out that the bisector of an angle of a triangle divides the opposite side not just in some way, 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 - exit to “space” - additional formation!
Let's go straight.
For what? We'll see now.
Let's continue the bisector until it intersects with the line.
Is this a familiar picture? Yes, yes, yes, exactly the same as in point 4, case 1 - it turns out that (- bisector)
Lying crosswise
So, that too.
Now let's look at the triangles and.
What can you say about them?
They are similar. Well, yes, their angles are equal as vertical ones. So, in two corners.
Now we have the right to write the relations of the relevant parties.
And now in short notation:
Oh! Reminds me of something, right? Isn't this what we wanted to prove? Yes, yes, exactly that!
You see how great the “spacewalk” proved to be - the construction of an additional straight line - without it nothing would have happened! And so, we have proven that
Now you can safely use it! Let's look at one more property of the bisectors of the corners of a triangle - don't be alarmed, now the hardest part is over - it will be easier.
We get that
Theorem 1:
Theorem 2:
Theorem 3:
Theorem 4:
Theorem 5:
Theorem 6: