Intersection of bisectors in an isosceles triangle. Bisector of a triangle - what is it
The interior angles of a triangle is called the bisector of the triangle.
The angle bisector of a triangle is also understood as the segment between its vertex and the point of intersection of the bisector with the opposite side of the triangle.
Theorem 8.
The three bisectors of a triangle intersect at one point.
Indeed, consider first the point Р of the intersection of two bisectors, for example, AK 1 and VC 2. This point is equally distant from the sides AB and AC, since it lies on the bisector of angle A, and is equally distant from the sides AB and BC, as belonging to the bisector of angle B. Therefore, it is equally distant from the sides AC and BC and thus belongs to the third bisector SK 3 , that is, at the point P all three bisectors intersect.
Properties of bisectors of internal and external angles of a triangle
Theorem 9.
The bisector of the interior angle of a triangle divides the opposite side into parts proportional to the adjacent sides. 
Proof. Consider the triangle ABC and the bisector of its angle B. Let us draw a straight line CM through the vertex C, parallel to the bisector BK, until it intersects at the point M as an extension of the side AB. Since VC is the bisector of the angle ABC, then ∠ ABK=∠ KBC. Further, ∠ ABK=∠ VMS, as the corresponding angles at parallel lines, and ∠ KBC=∠ VCM, as the cross-lying angles at parallel lines. Hence ∠ VCM=∠ VMS, and therefore the VMS triangle is isosceles, hence BC=VM. According to the theorem on parallel lines intersecting the sides of an angle, we have AK:K C=AB:VM=AB:BC, which was required to be proved.
Theorem 10
Bisector outer corner In a triangle ABC has a similar property: the segments AL and CL from the vertex A and C to the point L of the intersection of the bisector with the extension of the AC side are proportional to the sides of the triangle: AL: CL=AB :BC .
This property is proved in the same way as the previous one: an auxiliary line CM is drawn in the figure, parallel to the bisector BL . The angles BMC and BCM are equal, which means that the sides BM and BC of the triangle BMC are equal. From which we come to the conclusion AL:CL=AB:BC.
Theorem d4.
(the first formula for the bisector): If in triangle ABC the segment AL is the bisector of angle A, then AL? = AB AC - LB LC.
Proof: Let M be the point of intersection of the line AL with the circle circumscribed about the triangle ABC (Fig. 41). The BAM angle is equal to the MAC angle by convention. Angles BMA and BCA are equal as inscribed angles based on the same chord. Hence, triangles BAM and LAC are similar in two angles. Therefore, AL: AC = AB: AM. So AL AM = AB AC<=>AL (AL + LM) = AB AC<=>AL? = AB AC - AL LM = AB AC - BL LC. Which is what needed to be proven. Note: for the theorem on segments of intersecting chords in a circle and on inscribed angles, see the topic circle and circle.
Theorem d5.
(second formula for the bisector): In triangle ABC with sides AB=a, AC=b and angle A equal to 2? and the bisector l, the equality takes place:
l = (2ab / (a+b)) · cos?.
Proof: Let ABC be a given triangle, AL its bisector (Fig. 42), a=AB, b=AC, l=AL. Then S ABC = S ALB + S ALC . Hence absin2? = alsin? +blsin?<=>2absin? cos? = (a + b)lsin?<=>l = 2 (ab / (a+b)) cos?. The theorem has been proven.
The bisector of a triangle is a common geometric concept that does not cause much difficulty in learning. Knowing about 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 placed on the side that is located opposite it, while dividing the space into two identical parts.
Many teachers for quick associative memorization of mathematical concepts by students use different terminology, which is displayed in verses or associations. Of course, this definition is recommended for older children.
How is this line marked? Here we rely on the rules for designating 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 the point of intersection with the opposite side of the vertex. Moreover, the beginning of the designation is written exactly from the top.
Attention! How many bisectors does a triangle have? The answer is obvious: as many as there are vertices - three.
Properties
In addition to the definition, there are not so many properties of this geometric concept in a school textbook. The first property of the bisector of a triangle, which schoolchildren are introduced to, is the inscribed center, and the second, directly related to it, is the proportionality of the segments. The bottom line is this:
- Whatever the dividing line, there are points on it that are on the same distance from the parties, which make up the space between the rays.
- 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.
- Parts of the side of a triangular geometric figure, into which it is divided by a dividing line, are in proportion to the sides forming the angle.
We will try to bring the rest of the features into a system and present additional facts that will help to better understand the merits of this geometric concept.
Length
One of the types of tasks that cause difficulty for schoolchildren is finding the length of the bisector of the angle of a triangle. The first option, in which its length is located, contains the following data:
- the size of the space between the rays, from the top of which the given segment emerges;
- 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 that make up the angle, by the cosine of its half, to the sum of the sides.
Let's look at a specific example. Suppose we are given a figure ABC, in which the segment is drawn from angle A and intersects side BC at point K. We denote the value of A by Y. Based on this, AK \u003d (2 * AB * AC * cos (Y / 2)) / (AB + AS).
The second version of the problem, in which the length of the bisector of a triangle is determined, contains the following data:
- the values of all sides of the figure are known.
When solving a problem of this type, initially determine the semiperimeter. To do this, add the values of all sides and divide in half: p \u003d (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 twice root of the second degree from the product of the lengths of the sides that are adjacent to the top, to the semi-perimeter and the difference between the semi-perimeter and the length of the opposite side to the sum of the sides that make up the angle. That is, AK \u003d (2٦AB * AC * p * (r-BC)) / (AB + AC).
Attention! To make it easier to master the material, you can refer to comic tales available on the Internet that tell about the "adventures" of this line.
Instruction
If a given triangle is isosceles or regular, that is, it has
two or three sides, then its bisector, according to the property triangle, will also be the median. And, therefore, the opposite will divide the bisector in half.
Measure the opposite side with a ruler triangle where the bisector will tend to. Divide this side in half and put a dot in the middle of the side.
Draw a straight line through the constructed point and the opposite vertex. This will be the bisector triangle.
Sources:
- Medians, bisectors and heights of a triangle
Dividing an angle in half and calculating the length of a line drawn from its top to the opposite side is necessary for cutters, surveyors, fitters and people of some other professions.
You will need
- Tools Pencil Ruler Protractor Tables of sines and cosines Mathematical formulas and concepts: Definition of a bisector Sine and cosine theorems Bisector theorem
Instruction
Build a triangle of the necessary and magnitude, depending on what you are given? dfe sides and the angle between them, three sides or two angles and the side located between them.
Designate the vertices of the corners and the sides with traditional Latin A, B and C. The vertices of the corners are denoted, the opposite sides are lowercase. Label the corners Greek letters?,? And?
Using the sine and cosine theorems, calculate the angles and sides triangle.
Remember bisectors. Bisector - dividing the angle in half. Angle bisector triangle divides the opposite into two segments, which is equal to the ratio of the two adjacent sides triangle.
Draw the angle bisectors. Designate the resulting segments by the names of the angles, written in lowercase letters, with a subscript l. Side c is divided into segments a and b with indices l.
Calculate the lengths of the resulting segments using the sine theorem.
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note
The length of the segment, which is simultaneously the side of the triangle formed by one of the sides of the original triangle, the bisector and the segment itself, is calculated using the sine theorem. In order to calculate the length of another segment of the same side, use the ratio of the resulting segments and the adjacent sides of the original triangle.
In order not to get confused, draw the bisectors different angles different color.
bisector angle called a ray that starts at a vertex angle and divides it into two equal parts. Those. to spend bisector, you need to find the middle angle. The easiest way to do this is with a compass. In this case, you do not need to do any calculations, and the result will not depend on whether the value is angle whole number.
You will need
- compass, pencil, ruler.
Instruction
Leaving the width of the compass opening the same, set the needle at the end of the segment on one of the sides and draw a part of the circle so that it is located inside angle. Do the same with the second one. You will get two parts of the circles that will intersect inside angle- approximately in the middle. The parts of the circles can intersect at one or two points.
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Helpful advice
You can use a protractor to construct the angle bisector, but this method requires more precision. In this case, if the angle value is not an integer, the probability of errors in the construction of the bisector increases.
When building or developing home design projects, it is often necessary to build corner equal to the one already present. Templates come to the rescue school knowledge geometry.
Instruction
An angle is formed by two straight lines emanating from the same point. This point will be called the vertex of the corner, and the lines will be the sides of the corner.
Use three to indicate corners: one at the top, two at the sides. are called corner, starting with the letter that stands at one side, then they call the letter at the top, and then the letter at the other side. Use others to mark corners if you prefer otherwise. Sometimes only one letter is called, which is at the top. And you can denote the angles with Greek letters, for example, α, β, γ.
There are situations where it is necessary corner so that it is already given corner. If it is not possible to use a protractor when building, you can only get by with a ruler and a compass. Suppose, on the line marked with the letters MN, you need to build corner at point K, so that it is equal to angle B. That is, from point K it is necessary to draw a straight line, with line MN corner, which will be equal to angle B.
First, mark a point on each side of this corner, for example, points A and C, then connect points C and A with a straight line. Get tre corner nik ABC.
Now build on the line MN the same three corner vertex B is on the line at point K. Use the rule for constructing a triangle corner three o'clock. Set aside the segment KL from point K. It must be equal to the segment BC. Get point L.
From point K, draw a circle with a radius equal to the segment BA. From L draw a circle with radius CA. Connect the resulting point (P) of the intersection of two circles with K. Get three corner nick KPL, which will be equal to three corner niku ABC. So you get corner K. It will be equal to angle B. To make it more convenient and faster, set aside equal segments from vertex B, using one compass solution, without moving the legs, describe a circle with the same radius from point K.
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Tip 5: How to draw a triangle given two sides and a median
The triangle is the simplest geometric figure, which has three vertices connected in pairs by segments that form the sides of this polygon. The line segment connecting the vertex with the midpoint of the opposite side is called the median. Knowing the lengths of the two sides and the median connecting at one of the vertices, you can build a triangle without knowing the length of the third side or the angles.
Instruction
Draw a segment from point A, the length of which is one of the known sides of the triangle (a). Mark the end point of this segment with the letter B. After that, one of the sides (AB) of the desired triangle can already be considered built.
Use a compass to draw a circle with a radius equal to twice the length of the median (2∗m) and centered at point A.
Use the compass to draw a second circle with a radius equal to the length of the known side (b) and centered at point B. Put the compass aside for a while, but leave the measured one on it - you will need it again a little later.
Construct a line segment connecting point A with the intersection point of the two drawn by you. Half of this segment will be the one you are building - measure this half and put a point M. At this point, you have one side of the desired triangle (AB) and its median (AM).
Use a compass to draw a circle with a radius equal to the length of the second known side (b) and centered at point A.
Draw a segment that should start at point B, pass through point M and end at the point of intersection of the line with the circle you drew in the previous step. Designate the intersection point with the letter C. Now, in the required side, the side BC, unknown by the conditions of the problem, is also built.
The ability to divide any angle with a bisector is necessary not only in order to get an "A" in mathematics. This knowledge will be very useful to the builder, designer, surveyor and dressmaker. There are many things in life that need to be divided.
Everyone at school taught a joke about a rat that runs around the corners and divides the corner in half. This nimble and intelligent rodent was called the Bisector. It is not known how the rat divided the corner, and mathematicians in the school textbook "Geometry" can offer the following methods.
With the help of a protractor
The easiest way to draw a bisector is using a device for. It is necessary to attach the protractor to one side of the angle, aligning the reference point with its tip O. Then measure the angle in degrees or radians and divide it by two. With the help of the same protractor, set aside the degrees obtained from one of the sides and draw a straight line, which will become the bisector, to the point where angle O begins.With the help of a circle
You need to take a compass and breed it to any arbitrary size (within the drawing). Having set the tip at the point of the beginning of the angle O, draw an arc that intersects the rays, marking two points on them. Designate them A1 and A2. Then, setting the compass alternately at these points, two circles of the same arbitrary diameter should be drawn (on the scale of the drawing). The points of their intersection are designated C and B. Next, you need to draw a straight line through the points O, C and B, which will be the desired bisector.With a ruler
In order to draw the bisector of an angle with a ruler, you need to set aside segments of the same length from point O on the rays (sides) and designate them with points A and B. Then you should connect them with a straight line and use a ruler to divide the resulting segment in half, marking point C. The bisector is obtained by drawing a straight line through points C and O.Without tools
If there are no measuring tools, you can use ingenuity. It is enough just to draw an angle on tracing paper or ordinary thin paper and carefully fold the sheet so that the rays of the angle are aligned. The fold line in the drawing will be the desired bisector.Expanded angle
An angle greater than 180 degrees can be divided by a bisector in the same way. Only it will not be necessary to divide it, but the acute angle adjacent to it, remaining from the circle. The continuation of the found bisector will become the desired straight line, dividing the expanded angle in half.Angles in a triangle
It should be remembered that in an equilateral triangle, the bisector is also the median and height. Therefore, the bisector in it can be found by simply lowering the perpendicular to the side opposite from the angle (height) or dividing this side in half and connecting the midpoint with opposite corner(median).Related videos
The mnemonic rule “a bisector is a rat that runs around the corners and divides them in half” describes the essence of the concept, but does not give recommendations for constructing a bisector. To draw it, in addition to the rule, you will need a compass and a ruler.
Instruction
Let's say you need to build bisector corner A. Take a compass, put it with a point at point A (angle) and draw a circle of any . Where it intersects the sides of the corner, place points B and C.
Measure the radius of the first circle. Draw another one with the same radius, placing the compass at point B.
Draw the next circle (equal in size to the previous ones) centered at point C.
All three circles must intersect at one point - let's call it F. Using a ruler, draw a ray passing through points A and F. This will be the desired bisector of angle A.
There are several rules to help you find. For example, it is opposite in, equal to the ratio two adjacent sides. in isosceles
Today is going to be a very easy lesson. We will consider only one object - the angle bisector - and prove its most important property, which will be very useful to us in the future.
Just don’t relax: sometimes students who want to get a high score on the same OGE or USE, in the first lesson, cannot even formulate the exact definition of the bisector.
And instead of doing really interesting tasks, we spend time on such simple things. So read, watch - and adopt. :)
To begin with, a slightly 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: acute, obtuse and right 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, they say, in front of us is the angle $AOB$ (written as $\angle AOB$).
The captain seems to hint that in addition to the rays $OA$ and $OB$, one can always draw a bunch of rays from the point $O$. But among them there will be one special one - it is called the bisector.
Definition. The bisector of an angle is a ray that comes out of the vertex 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 angle
Since in real drawings it is far from always obvious that a certain ray (in our case, this is the $OM$ ray) splits the initial angle into two equal ones, it is customary in geometry 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 straight).
Okay, we figured out the definition. Now you need to understand what properties the bisector has.
Basic property of the angle bisector
In fact, the bisector has a lot of properties. And we will definitely consider 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 the given angle.
Translated from mathematical into Russian, this means two facts at once:
- Every point lying on the bisector of an angle is at the same distance from the sides of that 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? The good old definition of the distance from a point to a line will help us here:
Definition. The distance from a point to a line is the length of the perpendicular drawn from that point to that line.
For example, consider a line $l$ and a point $A$ not lying 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 line $l$.
Graphical representation of the distance from a point to a line Since an angle is just two rays, and each ray is a piece of a line, it's easy to determine the distance from a point to the sides of an angle. It's just two perpendiculars:
Determine the distance from a point to the sides of an angle That's all! Now we know what distance is and what a bisector is. Therefore, we can prove the main property.
As promised, we break the proof into two parts:
1. The 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 same point $M$ is at the same distance from the sides of the angle.
Proof. Let's draw perpendiculars from the point $M$ to the sides of the angle. 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 assumption (since $OM$ is a bisector);
- $\angle M((H)_(1))O=\angle M((H)_(2))O=90()^\circ $ by construction;
- $\angle OM((H)_(1))=\angle OM((H)_(2))=90()^\circ -\angle MO((H)_(1))$ because the sum acute angles of a right triangle is always equal to 90 degrees.
Therefore, 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 angle 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, i.e. $\angle MO((H)_(1))=\angle MO((H)_(2))$.
Proof. To begin with, let's draw this very ray $OM$, otherwise there will be nothing to prove:
Spent the beam $OM$ inside the corner
We got two right triangles again: $\vartriangle OM((H)_(1))$ and $\vartriangle OM((H)_(2))$. Obviously they are equal because:
- The hypotenuse $OM$ is common;
- The legs $M((H)_(1))=M((H)_(2))$ by condition (because 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, triangles $\vartriangle OM((H)_(1))$ and $\vartriangle OM((H)_(2))$ 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 formed equal angles with red arcs:
The bisector split the angle $\angle ((H)_(1))O((H)_(2))$ 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 on to new level. In the next lesson, we'll go over more complex properties bisectors and learn how to apply them to solve real problems.
PROPERTIES OF THE BISSECTOR
Bisector property: In a triangle, the bisector divides the opposite side into segments proportional to the adjacent sides.
Bisector of an external angle The bisector of an external angle of a triangle intersects the extension of its side at a point, the distances from which to the ends of this side are proportional, respectively, to the adjacent sides of the triangle. C B A D
Bisector length formulas:
The formula for finding the lengths of the segments into which the bisector divides the opposite side of the triangle
The formula for finding the ratio of the lengths of the segments into which the bisector is divided by the intersection point of the bisectors
Problem 1. One of the bisectors of a triangle is divided by the intersection point of the bisectors in a ratio of 3:2, counting from the vertex. Find the perimeter of a triangle if the length of the side of the triangle to which this bisector is drawn is 12 cm.
Solution We use the formula to find the ratio of the lengths of the segments into which the bisector is divided by the intersection point of the bisectors in the triangle: 30. Answer: P = 30cm.
Task 2 . Bisectors BD and CE ∆ ABC intersect at point O. AB=14, BC=6, AC=10. Find O D .
Solution. Let's use the formula for finding the length of the bisector: We have: BD = BD = = According to the formula for the ratio of the segments into which the bisector is divided by the intersection point of the bisectors: l = . 2 + 1 = 3 parts of everything.
this is part 1 OD = Answer: OD =
Problems In ∆ ABC, the bisectors AL and BK are drawn. Find the length of the segment KLif AB \u003d 15, AK \u003d 7.5, BL \u003d 5. In ∆ ABC, the bisector AD is drawn, and through point D is a straight line parallel to AC and intersecting AB at point E. Find the ratio of areas ∆ ABC and ∆ BDE , if AB = 5, AC = 7. Find the bisectors of acute angles of a right triangle with legs 24 cm and 18 cm. IN right triangle the bisector of an acute angle divides the opposite leg into segments 4 and 5 cm long. Determine the area of the triangle.
5. In an isosceles triangle, the base and side are 5 and 20 cm, respectively. Find the bisector of the angle at the base of the triangle. 6. Find the bisector of the right angle of a triangle whose legs are equal a and b. 7. Calculate the length of the bisector of angle A of triangle ABC with side lengths a = 18 cm, b = 15 cm, c = 12 cm. Find the ratio in which the bisectors divide internal corners at the point of their intersection.
Answers: Answer: Answer: Answer: Answer: Answer: Answer: Answer: Answer: AP = 6 AP = 10 see KL = CP =
