The median is the segment drawn from the vertex of the triangle to the middle of the opposite side, that is, it divides it in half by the point of intersection. The point at which the median intersects the side opposite to the top from which it emerges is called the base. Through one point, called the point of intersection, passes each median of the triangle. The formula for its length can be expressed in several ways.

Formulas for expressing the length of the median

• Often in problems in geometry, students have to deal with such a segment as the median of a triangle. The formula for its length is expressed in terms of the sides:

where a, b and c are sides. Moreover, c is the side on which the median falls. This is how the simplest formula looks like. Triangle medians are sometimes required for auxiliary calculations. There are other formulas as well.

• If during the calculation two sides of the triangle and a certain angle α located between them are known, then the length of the median of the triangle, lowered to the third side, will be expressed as follows.

Basic properties

• All medians have one common point of intersection O and they are also divided by it in a ratio of two to one, if we count from the top. This point is called the center of gravity of the triangle.
• The median divides the triangle into two others, the areas of which are equal. Such triangles are called equal triangles.
• If you draw all the medians, then the triangle will be divided into 6 equal figures, which will also be triangles.
• If in a triangle all three sides are equal, then in it each of the medians will also be a height and a bisector, that is, perpendicular to the side to which it is drawn, and bisects the angle from which it exits.
• In an isosceles triangle, the median dropped from a vertex that is opposite a side that is not equal to any other will also be the height and the bisector. Medians dropped from other vertices are equal. This is also a necessary and sufficient condition for isosceles.
• If the triangle is the base of a regular pyramid, then the height lowered onto this base is projected to the intersection point of all medians.

• In a right triangle, the median drawn to the longest side is half its length.
• Let O be the point of intersection of the medians of the triangle. The formula below will be true for any point M.

• Another property is the median of a triangle. The formula for the square of its length in terms of the squares of the sides is presented below.

Properties of the sides to which the median is drawn

• If we connect any two points of intersection of the medians with the sides on which they are lowered, then the resulting segment will be the midline of the triangle and be one half of the side of the triangle with which it has no common points.
• The bases of the heights and medians in the triangle, as well as the midpoints of the segments connecting the vertices of the triangle with the point of intersection of the heights, lie on the same circle.

In conclusion, it is logical to say that one of the most important segments is precisely the median of the triangle. Its formula can be used to find the lengths of its other sides.

To find the median on the sides of a triangle, it is not necessary to memorize an additional formula. It is enough to know the solution algorithm.

First, let's look at the problem in general terms.

Given a triangle with sides a, b, c. Find the length of the median drawn to side b.

AB=a, AC=b, BC=c.

On the ray BF we set aside the segment FD, FD=BF.

Let's connect point D with points A and C.

Quadrilateral ABCD is a parallelogram (by feature), since its diagonals at the intersection point are divided in half.

Property of the diagonals of a parallelogram: the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Hence: AC²+BD²=2(AB²+BC²), so b²+BD²=2(a²+c²),

BD²=2(a²+c²)-b². By construction, BF is half of BD, therefore,

This is the formula for finding the median of a triangle along its sides. It is usually written like this:

Let's move on to a specific problem.

The sides of the triangle are 13 cm, 14 cm and 15 cm. Find the median of the triangle drawn to its mid-length side.

Applying similar reasoning, we get:

AC²+BD²=2(AB²+BC²).

14²+BD²=2(13²+15²)

A triangle is 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 one straight line (see Fig. 1).

Basic elements of triangle abc

Peaks – points A, B, and C;

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

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

The angle formed by the sides of the triangle and lying in its interior is called the interior angle, and the angle 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 are also considered that have interesting properties: heights, medians, bisectors and midlines.

Height

Heights of a triangle are the perpendiculars dropped from the vertices of the triangle to opposite sides.

To build the height, do the following:

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

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

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

Triangle height properties

In a right triangle, the height drawn from the vertex of the right angle divides it into two triangles similar to the original triangle.

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

If the triangle is acute-angled, then all the bases of the heights belong to the sides of the triangle, and for an obtuse triangle, two heights fall on the extension of the sides.

Three heights in an 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 build a median, do the following:

1) find the middle of the side;

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

Triangle median properties

The median divides the triangle into two triangles of the same area.

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

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

Bisector

bisectors(from lat. bis - twice "and seko - I cut) call the segments of straight lines enclosed inside the triangle that bisect its corners (see Fig. 4).

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

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

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

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

Triangle bisector properties

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

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

The bisectors of the inner and outer corners are perpendicular.

If the bisector of the outer angle of the triangle intersects the continuation of the opposite side, then ADBD=ACBC.

The bisectors of one interior and two exterior angles of a triangle intersect at one point. This point is the center of one of the three excircles of this triangle.

The bases of the bisectors of two internal and one external angles of a triangle lie on the same 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 line.

When studying any topic of the school course, you can select a certain minimum of tasks, having mastered the methods for solving which, students will be able to solve any task at the level of program requirements for the topic being studied. I propose to consider tasks that will allow you to see the relationship between individual topics of the school mathematics course. Therefore, the compiled system of tasks is an effective means of repeating, summarizing and systematizing educational material in the course of preparing students for the exam.

To pass the exam, additional information about some elements of the triangle will not be superfluous. Consider the properties of the median of a triangle and problems in which these properties can be used. The proposed tasks implement the principle of level differentiation. All tasks are conditionally divided into levels (the level is indicated in brackets after each task).

Recall some properties of the median of a triangle

Property 1. Prove that the median of the triangle ABC drawn from the top A, less than half the sum of the sides AB and AC.

Proof

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Property 2. The median cuts the triangle into two equal areas.

Proof

From vertex B of triangle ABC, draw median BD and height BE..gif" alt="(!LANG:Area" width="82" height="46">!}

Since segment BD is a median, then

Q.E.D.

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Proof

Let us prove that the area of ​​each of the six triangles into which the medians divide triangle ABC is equal to the area of ​​triangle ABC. To do this, consider, for example, the triangle AOF and drop the perpendicular AK from the vertex A to the line BF .

Due to property 2,

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Property 6. The median of a right triangle drawn from the vertex of the right angle is half the hypotenuse.

Proof

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Consequences:1. The center of a circle circumscribed about a right triangle lies at the midpoint of the hypotenuse.

2. If in a triangle the length of the median is equal to half the length of the side to which it is drawn, then this triangle is a right triangle.

TASKS

When solving each subsequent problem, proven properties are used.

№1 Topics: Doubling the median. Difficulty: 2+

Features and properties of a parallelogram Classes: 8,9

Condition

On the continuation of the median AM triangle ABC per point M segment postponed MD, equal to AM. Prove that the quadrilateral ABDC- parallelogram.

Decision

Let's use one of the signs of a parallelogram. Diagonals of a quadrilateral ABDC intersect at a point M and divide it in half, so the quadrilateral ABDC- parallelogram.