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

2. 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.

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

Triangle bisector properties

1. The bisector of an angle is the locus of points equidistant from the sides of this angle.

2. The bisector of the interior angle of a triangle divides the opposite side into segments proportional to the adjacent sides: .

3. The point of intersection of the bisectors of a triangle is the center of a circle inscribed in this triangle.

Triangle height properties

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

2. In an acute triangle, two of its heights cut off similar triangles.

Properties of the perpendicular bisectors of a triangle

1. Each point of the perpendicular bisector to a segment is equidistant from the ends of this segment. The converse statement is also true: each point equidistant from the ends of the segment lies on the perpendicular bisector to it.

2. The point of intersection of the midperpendiculars drawn to the sides of the triangle is the center of the circle circumscribed about this triangle.

Property of the midline of a triangle

The midline of a triangle is parallel to one of its sides and equal to half of that side.

similarity of triangles

Two triangles are similar if one of the following conditions is met, called signs of similarity:

two angles of one triangle are equal to two angles of another triangle;

two sides of one triangle are proportional to two sides of another triangle, and the angles formed by these sides are equal;

The three sides of one triangle are respectively proportional to the three sides of the other triangle.

In similar triangles, the corresponding lines (heights, medians, bisectors, etc.) are proportional.

Sine theorem

Cosine theorem

a 2= b 2+ c 2- 2bc cos

Triangle area formulas

1. Arbitrary triangle

a, b, c - sides; - angle between sides a and b; - semiperimeter; R- radius of the circumscribed circle; r- radius of the inscribed circle; S- square; h a - height drawn to side a.

S = ah a

S = ab sin

S = pr

2. Right triangle

a, b- legs; c- hypotenuse; hc - height to side c.

S = ch c S = ab

3. Equilateral triangle

Quadrangles

Parallelogram Properties

Opposite sides are equal

Opposite angles are equal

the diagonals of the intersection point are divided in half;

the sum of the angles adjacent to one side is 180°;

The sum of the squares of the diagonals is equal to the sum of the squares of all sides:

d 1 2 +d 2 2 =2(a 2 +b 2).

A quadrilateral is a parallelogram if:

1. Its two opposite sides are equal and parallel.

2. Opposite sides are equal in pairs.

3. Opposite angles are equal in pairs.

4. The diagonals of the intersection point are divided in half.

Trapezoid Properties

Its midline is parallel to the bases and equal to their half-sum;

if the trapezoid is isosceles, then its diagonals are equal and the angles at the base are equal;

if the trapezoid is isosceles, then a circle can be circumscribed around it;

If the sum of the bases is equal to the sum of the sides, then a circle can be inscribed in it.

Rectangle properties

the diagonals are equal.

A parallelogram is a rectangle if:

1. One of its corners is right.

2. Its diagonals are equal.

Rhombus Properties

all the properties of a parallelogram;

Diagonals are perpendicular

the diagonals are the bisectors of its angles.

1. A parallelogram is a rhombus if:

2. Its two adjacent sides are equal.

3. Its diagonals are perpendicular.

4. One of the diagonals is the bisector of its angle.

Square properties

All corners of the square are right

The diagonals of the square are equal, mutually perpendicular, the intersection point is divided in half and the corners of the square are divided in half.

A rectangle is a square if it has some characteristic of a rhombus.

Basic Formulas

1. Arbitrary convex quadrilateral
d1,d2- diagonals; - the angle between them; S- square.

S=d 1 d 2 sin

Note. AT this lesson outlined theoretical materials and solving problems in geometry on the topic "median in a right triangle". If you need to solve a problem in geometry, which is not here - write about it in the forum. Almost certainly the course will be expanded.

median properties right triangle

Definition of the median

  • The medians of a triangle intersect at one point and are divided by this point into two parts in a ratio of 2:1, counting from the top of the angle. The point of their intersection is called the center of gravity of the triangle (the term "centroid" is used relatively rarely in problems to designate this point),
  • The median divides the triangle into two triangles of equal area.
  • A triangle is divided by three medians into six triangles of equal area.
  • The longer side of the triangle corresponds to the smaller median.

Geometry problems proposed for solution mainly use the following median properties of a right triangle.

  • The sum of the squares of the medians dropped on the legs of a right triangle is equal to five squares of the median dropped on the hypotenuse (Formula 1)
  • Median dropped to the hypotenuse of a right triangle equal to half the hypotenuse(Formula 2)
  • Median dropped to the hypotenuse of a right triangle equal to the radius of the circle circumscribed around given right triangle (Formula 2)
  • Median dropped to the hypotenuse equal to half the square root of the sum of the squares of the legs(Formula 3)
  • The median dropped to the hypotenuse is equal to the quotient of dividing the length of the leg by two sines of the opposite leg acute angle(Formula 4)
  • The median dropped to the hypotenuse is equal to the quotient of dividing the length of the leg by two cosines of the acute angle adjacent to the leg (Formula 4)
  • The sum of the squares of the sides of a right triangle is equal to eight squares of the median dropped to its hypotenuse (Formula 5)

Symbols in formulas:

a, b- legs of a right triangle

c- the hypotenuse of a right triangle

If we denote the triangle as ABC, then

Sun = a

(that is sides a,b,c- are opposite to the corresponding angles)

m a- median drawn to leg a

m b- median drawn to leg b

m c - median of a right triangle drawn to the hypotenuse with

α (alpha)- angle CAB opposite side a

Problem about the median in a right triangle

The medians of a right triangle drawn to the legs are 3 cm and 4 cm, respectively. Find the triangle's hypotenuse

Solution

Before starting to solve the problem, let's pay attention to the ratio of the length of the hypotenuse of a right triangle and the median that is lowered onto it. To do this, we turn to formulas 2, 4, 5 median properties in a right triangle. These formulas explicitly indicate the ratio of the hypotenuse and the median, which is lowered to it as 1 to 2. Therefore, for the convenience of future calculations (which will not affect the correctness of the solution in any way, but will make it more convenient), we denote the lengths of the legs AC and BC through the variables x and y as 2x and 2y (not x and y).

Consider a right triangle ADC. Angle C is a straight line according to the condition of the problem, leg AC is common with triangle ABC, and leg CD is equal to half of BC according to the properties of the median. Then, by the Pythagorean theorem

AC 2 + CD 2 = AD 2

Since AC \u003d 2x, CD \u003d y (since the median divides the leg into two equal parts), then
4x2 + y2 = 9

At the same time, consider a right triangle EBC. It also has a right angle C by the condition of the problem, the leg BC is common with the leg BC of the original triangle ABC, and the leg EC by the property of the median is equal to half of the leg AC of the original triangle ABC.
According to the Pythagorean theorem:
EC 2 + BC 2 = BE 2

Since EC \u003d x (the median bisects the leg), BC \u003d 2y, then
x2 + 4y2 = 16

Since triangles ABC, EBC and ADC are connected by common sides, both equations obtained are also connected.
Let's solve the resulting system of equations.
4x2 + y2 = 9
x2 + 4y2 = 16

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.

    Bisectors internal corners triangles 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 opposite sides, then their bases lie on the same straight 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 under study. 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 effective tool repetition, generalization and systematization educational material in preparing students for the exam.

To pass the exam will not be superfluous additional information about some elements of the triangle. 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.

Solution

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.