Instructions

If for given triangle isosceles or regular, that is, he has
two or three sides, then its bisector, according to the property triangle, will also be the median. And, therefore, the opposite one will be divided in half by the bisector.

Measure the opposite side with a ruler triangle, where the bisector will tend. Divide this side in half and place a dot in the middle of the side.

Draw a straight line passing through the constructed point and the opposite vertex. This will be the bisector triangle.

Sources:

• Medians, bisectors and altitudes of a triangle

Dividing an angle in half and calculating the length of a line drawn from its top to the opposite side is something that cutters, surveyors, installers and people of some other professions need to be able to do.

You will need

• Tools Pencil Ruler Protractor Sine and Cosine Tables Mathematical formulas and concepts: Definition of a bisector Theorems of sines and cosines Bisector theorem

Instructions

Construct a triangle of the required size, depending on what is given to you? dfe sides and the angle between them, three sides or two angles and the side located between them.

Label the vertices of the corners and sides with the traditional Latin letters A, B and C. The vertices of the corners are denoted by , and the opposite sides are denoted by lowercase letters. Label the corners Greek letters?,? And?

Using the theorems of sines and cosines, calculate the angles and sides triangle.

Remember bisectors. Bisector - dividing an angle in half. Angle bisector triangle divides the opposite into two segments, which are equal to the ratio of the two adjacent sides triangle.

Draw the bisectors of the angles. Label the resulting segments with the names of the angles written lowercase letters, with subscript l. Side c is divided into segments a and b with indices l.

Calculate the lengths of the resulting segments using the law of sines.

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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 law of sines. 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.

Helpful advice

To avoid confusion, draw bisectors different angles different colors.

Bisector angle called a ray that starts at the 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 quantity is angle an integer.

You will need

• compass, pencil, ruler.

Instructions

Leaving the width of the compass opening the same, place the needle at the end of the segment on one of the sides and draw part of the circle so that it is located inside angle. Do the same with the second one. You will end up with two parts of circles that will intersect inside angle- approximately in the middle. Parts of circles can intersect at one or two points.

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Helpful advice

To construct the bisector of an angle, you can use a protractor, but this method requires greater accuracy. Moreover, if the angle value is not an integer, the likelihood of errors in constructing the bisector increases.

When building or developing home design projects, it is often necessary to build corner, equal to what is already available. Templates come to the rescue school knowledge geometry.

Instructions

An angle is formed by two straight lines emanating from one point. This point will be called the vertex of the angle, and the lines will be the sides of the angle.

Use three to indicate corners: one at the top, two at the sides. Called corner, starting with the letter that stands on one side, then the letter that stands at the top is called, and then the letter on the other side. Use others to indicate angles if you prefer otherwise. Sometimes only one letter is named, which is at the top. And you can denote angles with Greek letters, for example, α, β, γ.

There are situations when it is necessary corner, so that it is narrower than the given angle. If it is not possible to use a protractor when constructing, you can only get by with a ruler and a compass. Suppose, on a straight line marked with the letters MN, you need to construct 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.

Start by marking a point on each side. given angle, for example, points A and C, then connect points C and A with a straight line. Get tre corner nik ABC.

Now build the same tre on the straight line MN corner so that its vertex B is on the line at point K. Use the rule for constructing a triangle corner nnik in three. Lay off the segment KL from point K. It must be equal to the segment BC. Get the L point.

From point K, draw a circle with a radius equal to segment BA. From L, draw a circle with radius CA. Connect the resulting point (P) of intersection of two circles with K. Get three corner KPL, which will be equal to three corner ABC book. This is how you get corner K. It will be equal to angle B. To make this more convenient and faster, set aside from vertex B equal segments, using one compass opening, without moving the legs, describe a circle with the same radius from point K.

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Tip 5: How to construct a triangle using two sides and a median

A triangle is the simplest geometric figure that has three vertices connected in pairs by segments that form the sides of this polygon. The segment connecting the vertex to the middle of the opposite side is called the median. Knowing the lengths of two sides and the median connecting at one of the vertices, you can construct a triangle without having information about the length of the third side or the size of the angles.

Instructions

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 this, one of the sides (AB) of the desired triangle can already be considered constructed.

Using a compass, draw a circle with a radius equal to twice the length of the median (2∗m) and with a center at point A.

Using a compass, draw a second circle with a radius equal to length known party(b), and with the center 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 to the intersection point of the two you drew. Half of this segment will be the one you are building - measure this half and put point M. At this moment you have one side of the desired triangle (AB) and its median (AM).

Using a compass, 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 straight line with the circle you drew in the previous step. Designate the point of intersection with the letter C. Now the side BC, unknown according to the conditions of the problem, has been constructed in the desired one.

The ability to divide any angle with a bisector is needed not only to get an “A” in mathematics. This knowledge will be very useful for builders, designers, surveyors and dressmakers. In life, you need to be able to divide many things in half.

Everyone at school learned a joke about a rat that runs around corners and divides the corner in half. The name of this nimble and intelligent rodent was Bisector. It is not known how the rat divided the corner, and mathematicians school textbook"Geometry" the following methods can be proposed.

Using a protractor

The easiest way to conduct a bisector is using a device for. You need 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. Using the same protractor, set aside the obtained degrees from one of the sides and draw a straight line, which will become a bisector, to the starting point of angle O.

Using a compass

You need to take a compass and move it to any arbitrary size (within the limits of the drawing). Having placed the tip at the starting point of angle O, draw an arc intersecting the rays, marking two points on them. They are designated A1 and A2. Then, placing the compass alternately at these points, you should draw two circles of the same arbitrary diameter (on the scale of the drawing). Their intersection points are designated C and B. Next, you need to draw a straight line through points O, C and B, which will be the desired bisector.

Using a ruler

In order to draw the bisector of an angle using a ruler, you need to plot segments from point O on the rays (sides) same length and designate them as points A and B. Then you should connect them with a straight line and, using a ruler, divide the resulting segment in half, designating point C. A bisector will be obtained if you draw a straight line through points C and O.

No tools

If not measuring instruments, you can use your ingenuity. It is enough to simply draw an angle on tracing paper or ordinary thin paper and carefully fold the piece of paper so that the rays of the angle align. The fold line in the drawing will be the desired bisector.

Straight angle

An angle greater than 180 degrees can be divided by a bisector using the same methods. Only it will be necessary to divide not 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 unfolded angle in half.

Angles in a triangle

It should be remembered that in 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 to the angle (height) or dividing this side in half and connecting the midpoint with opposite angle(median).

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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 provide recommendations for constructing a bisector. To draw it, in addition to the rule, you will need a compass and a ruler.

Instructions

Let's say you need to build bisector angle A. Take a compass, place its tip 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 a compass at point B.

Draw the next circle (equal in size to the previous ones) with its center 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 that will help you find. For example, it is the opposite in, equal to the ratio two adjacent sides. In isosceles

Triangle - a polygon with three sides, or a closed broken line with three links, or a figure formed by three segments connecting three points that do not lie on the same straight line (see Fig. 1).

Essential elements triangle abc

Peaks – points A, B, and C;

Parties – segments a = BC, b = AC and c = AB connecting the vertices;

Angles – α, β, γ formed by three pairs of sides. Angles are often designated in the same way as vertices, with the letters A, B, and C.

The angle formed by the sides of a triangle and lying in its interior area is called an interior angle, and the one adjacent to it is the adjacent angle of the triangle (2, p. 534).

Heights, medians, bisectors and midlines of a triangle

In addition to the main elements in a triangle, other segments with interesting properties are also considered: heights, medians, bisectors and midlines.

Height

Triangle heights- these are perpendiculars dropped from the vertices of the triangle to opposite sides.

To plot the height, you must perform the following steps:

1) draw a straight line containing one of the sides of the triangle (if the height is drawn from the vertex acute angle in an obtuse triangle);

2) from the vertex lying opposite the drawn line, draw a segment from the point to this line, making an angle of 90 degrees with it.

The point where the altitude intersects the side of the triangle is called height base (see Fig. 2).

Properties of triangle altitudes

In a right triangle, the altitude drawn from the vertex right angle, splits it into two triangles similar to the original triangle.

In an acute triangle, its two altitudes cut off similar triangles from it.

If the triangle is acute, then all the bases of the altitudes belong to the sides of the triangle, and obtuse triangle two heights fall on the continuation of the sides.

Three heights in acute triangle intersect at one point and this point is called orthocenter triangle.

Median

Medians(from Latin mediana – “middle”) - these are segments connecting the vertices of the triangle with the midpoints of the opposite sides (see Fig. 3).

To construct the median, you must perform the following steps:

1) find the middle of the side;

2) connect the point that is the middle of the side of the triangle with the opposite vertex with a segment.

Properties of triangle medians

The median divides a triangle into two triangles of equal area.

The medians of a triangle intersect at one point, which divides each of them in a ratio of 2:1, counting from the vertex. This point is called center of gravity triangle.

The entire triangle is divided by its medians into six equal triangles.

Bisector

Bisectors(from Latin bis - twice and seko - cut) are the straight line segments enclosed inside a triangle that bisect its angles (see Fig. 4).

To construct a bisector, you must perform the following steps:

1) construct a ray coming out from the vertex of the angle and dividing it into two equal parts (the bisector of the angle);

2) find the point of intersection of the bisector of the angle of the triangle with opposite side;

3) select a segment connecting the vertex of the triangle with the intersection point on the opposite side.

Properties of triangle bisectors

The bisector of an angle of a triangle divides the opposite side in a ratio equal to the ratio of the two adjacent sides.

The bisectors of the interior angles of a triangle intersect at one point. This point is called the center of the inscribed circle.

The bisectors of the internal and external angles are perpendicular.

If the bisector of an exterior angle of a triangle intersects the extension of the opposite side, then ADBD=ACBC.

Bisectors of one internal and two external corners triangles intersect at one point. This point is the center of one of the three excircles this triangle.

The bases of the bisectors of two internal and one external angles of a triangle lie on the same straight line if the bisector of the external angle is not parallel to the opposite side of the triangle.

If the bisectors of the external angles of a triangle are not parallel to opposite sides, then their bases lie on the same straight line.

The interior angles of a triangle are called the triangle bisector.
The bisector of an angle 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, let us first consider the point P of intersection of two bisectors, for example AK 1 and VK 2. This point is equally distant from the sides AB and AC, since it lies on the bisector of angle A, and equally distant from the sides AB and BC, as belonging to the bisector of angle B. This means that it is equally distant from the sides AC and BC and thereby belongs to the third bisector CK 3, that is, at point P all three bisectors intersect.
Properties of the bisectors of the internal and external angles of a triangle
Theorem 9. Bisector internal corner of a triangle divides the opposite side into parts proportional to the adjacent sides.
Proof. Let us consider triangle ABC and the bisector of its angle B. Let us 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 VC is the bisector of angle ABC, then ∠ ABC = ∠ KBC. Further, ∠ АВК=∠ ВСМ, as corresponding angles for parallel lines, and ∠ КВС=∠ ВСМ, as crosswise angles for parallel lines. Hence ∠ ВСМ=∠ ВМС, and therefore the triangle ВСМ is isosceles, hence ВС=ВМ. According to the theorem about parallel lines intersecting the sides of an angle, we have AK:K C=AB:VM=AB:BC, which is what needed to be proven.
Theorem 10 Bisector of external angle B triangle ABC has a similar property: the segments AL and CL from the 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: AL: C.L.=AB:BC.
This property is proved in the same way as the previous one: in the figure an auxiliary line SM is drawn parallel to the bisector BL. The angles BMC and BC 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. (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 line AL with the circle circumscribed about triangle ABC (Fig. 41). Angle BAM is equal to angle MAC by convention. Angles BMA and BCA are congruent as inscribed angles subtended by the same chord. This means that 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 about segments of intersecting chords in a circle and about inscribed angles, see the topic circle and circle.

Theorem d5. (second formula for the bisector): In a triangle ABC with sides AB=a, AC=b and angle A equal to 2? and bisector l, the equality holds:
l = (2ab / (a+b)) cos?.

Proof: Let ABC be the given triangle, AL its bisector (Fig. 42), a=AB, b=AC, l=AL. Then S ABC = S ALB + S ALC. Therefore, absin2? = alsin? +blsin?<=>2absin?·cos? = (a + b) lsin?<=>l = 2·(ab / (a+b))· cos?. The theorem has been proven.

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, and besides, they are usually asked not about an angle, but about a geometric figure. So the bisector of a triangle is a ray that connects the vertex of the triangle to the 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 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 there is the center of a circle inscribed in a 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 of the basic functions of the bisector, as well as its basic properties, is necessary for solving 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.

What is the bisector of an angle of a triangle? When answering this question, the famous rat running around 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 with scientific point From a perspective, the answer to this question should sound something like this: starting at the vertex of the angle and dividing the latter into two equal parts." In geometry, this figure is also perceived as a segment of the bisector before its intersection with the opposite side of the triangle. This is not a mistaken opinion. But what What else is known about the bisector of an angle, besides its definition?

Like anyone else locus points, it has its own signs. 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.