The bisector of a triangle is a segment that divides an angle of a 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 an internal and external angle intersect, an angle of 90 0 is obtained. 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 of 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 to an equilateral triangle, that is, a triangle in which all three sides are equal.
Example assignment
In triangle ABC: BR is the 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 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.
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Lesson topic
Angle bisector
Lesson Objectives
To enhance schoolchildren’s knowledge about the bisector of an angle and its properties;
Introduce new information about the bisector of an angle;
Expand students’ knowledge that the theorem about the properties of the bisector can be proven in different ways;
Develop logical thinking, interest in mathematical sciences, perseverance and the ability to analyze.
Lesson Objectives
Expand students’ knowledge about the bisector of an angle;
Strengthen the skills of constructing an angle bisector using drawing tools;
Get additional and interesting information on this topic;
Give information about the significance of the theorem in the development of mathematics;
Consolidate acquired knowledge by solving problems;
To cultivate perseverance, curiosity and a desire to study mathematical sciences.
Lesson Plan
1. Disclosure of the main topic of the lesson about the bisector of an angle;
2. Repetition of the material covered;
3. Interesting information about the bisector.
4. Historical background, Greek geometry.
5. Homework.
Angle bisector
Today's lesson we will devote to the topic of bisectors. Let's remember the definitions of a bisector.
A bisector is a locus of points equidistant from the sides of an angle.
To put it simply, a bisector is a line that divides an angle in half.
The bisector of an angle is a ray emerging from the vertex of the angle and dividing it into two other equal angles.
The word “bisector” translated from French means something that cuts an angle in half or equally divides it in half.
Bisector of a triangle
In addition to the bisector of an angle, there is also the bisector of a triangle, because a triangle contains as many as three angles, respectively, each triangle can have three different bisectors.
What is the bisector of a triangle? The bisector of a triangle is the segment of the angle bisector that connects its vertex in a triangle with a point on the opposite side.
The bisector triangle has certain unique properties. For example, it divides the opposite side into segments that are proportional to the other two sides.
As for a right triangle, its bisectors of acute angles, when intersecting, form an angle of exactly 45 degrees.
In addition, one should not forget such a property of the bisectors of a triangle, such as the fact that they intersect strictly in the center of the circle inscribed in the triangle.
Well, the most interesting thing is that for an isosceles triangle, the line drawn to the base will be the bisector, the median, and the height. Accordingly, the inverse rule is that if the median, height and bisector, which is drawn from one vertex of the triangle, coincide, then we have an isosceles triangle.
What properties can you remember of a right and isosceles triangle?
Construction of the bisector
The angle bisector is constructed using a protractor using its degree measure. To begin constructing the bisector, we take and divide the degree measure in half and, putting the degree measure of the half angle on one side of the vertex, and then the second half becomes the bisector of the given angle.
We take a given angle, which has a degree measure of ninety degrees, and using the bisector we obtain two constructed angles of 45 degrees.
A straight angle uses a bisector to divide the angle into 2 right angles. When constructing a bisector, an obtuse angle divides it into 2 acute angles.
From the definition of a bisector, we know that it is a ray that bisects an angle. To construct a bisector, this means you need to divide the angle in half.
Algorithm for constructing an angle bisector
1. First, draw a circle with the center at the vertex of the angle so that it intersects its sides.
3. Draw 2 circles with a radius so that they have an intersection point inside this angle.
4. Now we draw a ray from the vertex of the angle in such a way that it passes through the intersection point of these circles. This ray is the bisector of this angle.
Now let's try to prove that the resulting ray is the bisector of this angle. Let's take the example of two triangles that have one side in common, that is, a segment from the vertex to the point of intersection of the circles, which we obtained in 3p.
The 2nd pair of corresponding sides are the segments obtained in step 1 that go from the vertex of the angle to the points of intersection of the circle with its sides.
The third pair of corresponding sides are respectively the segments obtained in 1p. from the points of intersection of the circle, to the point of intersection of the circles, but obtained in 3p.
Therefore, 2 pairs of these segments are equal, since they are the radii of one or two circles, but with the same radius. It follows that the triangles are equal on all three sides. It is known that when triangles are equal, then their angles are equal. Therefore, at the vertex, the two new angles and the given angles according to the conditions of the problem are equal, therefore, the constructed ray will be a bisector.
Interesting information about the bisector
Did you know that there is a science called mnemonics, which translated from Greek means the art of memorization. And in order to better remember the definition of a bisector, there is a mnemonic rule according to which a bisector is a rat that runs around the corners and divides the corner in half.
Did you know that Archimedes also used the bisector theorem? He used it to divide the base into parts that are proportional to the sides in order to determine the length of the half sides of a twelve-gon, 24-gon, etc.
The legend of the angle bisector
The Tale of Two Angles and a Bisector, or the Formation of an Adjacent Angle.
One day two corners met in the same square. The oldest angle was about 130 degrees, and the youngest was only fifty. Since this is a fairy tale, let’s replace years with degrees. So they met and began to argue which of them was better and more important. The elder believed that priority was on his side, since he was older, wiser and had seen more in his lifetime in his 130°. The younger one, on the contrary, insisted that he was younger, therefore stronger and more resilient. And so that the dispute would not last forever, they decided to hold a tournament. Bisector found out about these competitions and decided to defeat her enemies at the same time and lead Geometry.
And now the long-awaited time has come for the tournament, where there were 2 Corners. At the moment when the battles were in full swing, Bisector appeared and decided to take part. But then the older Angle first entered the battle with the Bisector, then the younger one joined in, and victory still ended up on the side of the Bisector.
Theorem. The bisector of an interior angle of a triangle divides the opposite side into parts proportional to the adjacent sides.
Proof. Consider triangle ABC (Fig. 259) and the bisector of its angle B. Draw through vertex C a straight line CM, parallel to the bisector BC, until it intersects at point M with the continuation of side AB. Since BK is the bisector of angle ABC, then . Further, as corresponding angles for parallel lines, and as crosswise angles for parallel lines. Hence and therefore - isosceles, whence . By the theorem about parallel lines intersecting the sides of an angle, we have and in view we get , which is what we needed to prove.
The bisector of the external angle B of triangle ABC (Fig. 260) has a similar property: the segments AL and CL from vertices A and C to the point L of the intersection of the bisector with the continuation of side AC are proportional to the sides of the triangle:
This property is proven in the same way as the previous one: in Fig. 260 an auxiliary straight line SM is drawn parallel to the bisector BL. The reader himself will be convinced of the equality of the angles VMS and VSM, and therefore the sides VM and BC of the triangle VMS, after which the required proportion will be obtained immediately.
We can say that the bisector of an external angle divides the opposite side into parts proportional to the adjacent sides; you just need to agree to allow “external division” of the segment.
Point L, lying outside the segment AC (on its continuation), divides it externally in the relation if Thus, the bisectors of the angle of a triangle (internal and external) divide the opposite side (internal and external) into parts proportional to the adjacent sides.
Problem 1. The sides of the trapezoid are equal to 12 and 15, the bases are equal to 24 and 16. Find the sides of the triangle formed by the large base of the trapezoid and its extended sides.
Solution. In the notation of Fig. 261 we have a proportion for the segment that serves as a continuation of the lateral side, from which we easily find. In a similar way, we determine the second lateral side of the triangle. The third side coincides with the large base: .
Problem 2. The bases of the trapezoid are 6 and 15. What is the length of the segment parallel to the bases and dividing the sides in the ratio 1:2, counting from the vertices of the small base?
Solution. Let's turn to Fig. 262, depicting a trapezoid. Through the vertex C of the small base we draw a line parallel to the side AB, cutting off the parallelogram from the trapezoid. Since , then from here we find . Therefore, the entire unknown segment KL is equal to Note that to solve this problem we do not need to know the lateral sides of the trapezoid.
Problem 3. The bisector of the internal angle B of triangle ABC cuts side AC into segments at what distance from vertices A and C will the bisector of the external angle B intersect the extension AC?
Solution. Each of the bisectors of angle B divides AC in the same ratio, but one internally and the other externally. Let us denote by L the point of intersection of the continuation AC and the bisector of the external angle B. Since AK Let us denote the unknown distance AL by then and we will have a proportion The solution of which gives us the required distance
Complete the drawing yourself.
Exercises
1. A trapezoid with bases 8 and 18 is divided by straight lines parallel to the bases into six strips of equal width. Find the lengths of the straight segments dividing the trapezoid into strips.
2. The perimeter of the triangle is 32. The bisector of angle A divides side BC into parts equal to 5 and 3. Find the lengths of the sides of the triangle.
3. The base of an isosceles triangle is a, the side is b. Find the length of the segment connecting the intersection points of the bisectors of the corners of the base with the sides.
What is the bisector of an angle of a triangle? To this question, the well-known rat running around the corners and dividing the corner in half comes out of the mouth of some people." If the answer should be "humorous", then perhaps it is correct. But from a scientific point of view, the answer to this question should be: something like this: starting at the top of an angle and dividing the latter into two equal parts." In geometry, this figure is also perceived as a segment of the bisector until it intersects with the opposite side of the triangle. This is not a misconception. What else is known about the bisector of an angle, besides its definition?
Like any geometric locus of points, it has its own characteristics. The first of them is, rather, not even a sign, but a theorem, which can be briefly expressed as follows: “If the side opposite to it is divided into two parts by a bisector, then their ratio will correspond to the ratio of the sides of a large triangle.”
The second property that it has: the point of intersection of the bisectors of all angles is called the incenter.
The third sign: the bisectors of one internal and two external angles of a triangle intersect at the center of one of the three inscribed circles.
The fourth property of the angle bisector of a triangle is that if each of them is equal, then the latter is isosceles.
The fifth sign also concerns an isosceles triangle and is the main guideline for its recognition in a drawing by bisectors, namely: in an isosceles triangle it simultaneously serves as the median and altitude.
The angle bisector can be constructed using a compass and ruler:
The sixth rule states that it is impossible to construct a triangle using the latter only with the existing bisectors, just as it is impossible to construct in this way the doubling of a cube, the squaring of a circle and the trisection of an angle. Strictly speaking, these are all the properties of the angle bisector of a triangle.
If you carefully read the previous paragraph, then perhaps you were interested in one phrase. "What is trisection of an angle?" - you will probably ask. The trisector is a little similar to the bisector, but if you draw the latter, the angle will be divided into two equal parts, and when constructing a trisection, it will be divided into three. Naturally, the bisector of an angle is easier to remember, because trisection is not taught in school. But for the sake of completeness, I’ll tell you about it too.
A trisector, as I already said, cannot be constructed only with a compass and a ruler, but it can be created using Fujita’s rules and some curves: Pascal’s snails, quadratrixes, Nicomedes’ conchoids, conic sections,
Problems on trisection of an angle are quite simply solved using nevsis.
In geometry there is a theorem about angle trisectors. It is called Morley's theorem. She states that the intersection points of the trisectors of each angle located in the middle will be the vertices
A small black triangle inside a large one will always be equilateral. This theorem was discovered by British scientist Frank Morley in 1904.
Here's how much you can learn about dividing an angle: The trisector and bisector of an angle always require detailed explanations. But here were given many definitions that I had not yet disclosed: Pascal’s snail, Nicomedes’ conchoid, etc. Rest assured, there is much more to write about them.
Among the numerous subjects of secondary school there is one such as “geometry”. It is traditionally believed that the founders of this systematic science are the Greeks. Today, Greek geometry is called elementary, since it was she who began the study of the simplest forms: planes, straight lines, and triangles. We will focus our attention on the latter, or rather on the bisector of this figure. For those who have already forgotten, the bisector of a triangle is a segment of the bisector of one of the corners of the triangle, which divides it in half and connects the vertex with a point located on the opposite side.
The bisector of a triangle has a number of properties that you need to know when solving certain problems:
- The bisector of an angle is the locus of points located at equal distances from the sides adjacent to the angle.
- The bisector in a triangle divides the side opposite the angle into segments that are proportional to the adjacent sides. For example, given a triangle MKB, where a bisector emerges from angle K, connecting the vertex of this angle with point A on the opposite side MB. Having analyzed this property and our triangle, we have MA/AB=MK/KB.
- The point at which the bisectors of all three angles of a triangle intersect is the center of a circle that is inscribed in the same triangle.
- The bases of the bisectors of one external and two internal angles are on the same straight line, provided that the bisector of the external angle is not parallel to the opposite side of the triangle.
- If two bisectors of one then this
It should be noted that if three bisectors are given, then constructing a triangle from them, even with the help of a compass, is impossible.
Very often, when solving problems, the bisector of a triangle is unknown, but it is necessary to determine its length. To solve this problem, you need to know the angle that is bisected by the bisector and the sides adjacent to this angle. In this case, the required length is defined as the ratio of twice the product of the sides adjacent to the corner and the cosine of the angle divided in half to the sum of the sides adjacent to the corner. For example, given the same triangle MKB. The bisector emerges from angle K and intersects the opposite side of the MV at point A. The angle from which the bisector emerges is denoted by y. Now let’s write down everything that is said in words in the form of a formula: KA = (2*MK*KB*cos y/2) / (MK+KB).
If the value of the angle from which the bisector of a triangle emerges is unknown, but all its sides are known, then to calculate the length of the bisector we will use an additional variable, which we will call the semi-perimeter and denote by the letter P: P=1/2*(MK+KB+MB). After this, we will make some changes to the previous formula by which the length of the bisector was determined, namely, in the numerator of the fraction we put double the product of the lengths of the sides adjacent to the corner by the semi-perimeter and the quotient, where the length of the third side is subtracted from the semi-perimeter. We will leave the denominator unchanged. In the form of a formula, it will look like this: KA=2*√(MK*KB*P*(P-MB)) / (MK+KB).
The bisector of an isosceles triangle, along with general properties, also has several of its own. Let's remember what kind of triangle this is. Such a triangle has two equal sides and equal angles adjacent to the base. It follows that the bisectors that fall on the lateral sides of an isosceles triangle are equal to each other. In addition, the bisector lowered to the base is both the height and the median.