The middle lines of the triangle are from the night. Thales's theorem

1 Additional construction leading to the triangle midline theorem, trapezoid and similarity properties of triangles.

And she equal to half the hypotenuse.
Corollary 1.
Corollary 2.

2 All right triangles with the same acute angle are similar. A look at trigonometric functions.

3 An example of an additional construction is a height lowered to the hypotenuse. Derivation of the Pythagorean theorem based on the similarity of triangles.

From this it is clear that

1 All right triangles with the same acute angle are similar. A look at trigonometric functions.

Triangles with hatched and non-hatched sides are similar in that their two angles are equal. Therefore where

This means that the indicated relations depend only on the acute angle of the right triangle and essentially determine it. This is one of the reasons for the appearance of trigonometric functions:

Often writing trigonometric functions of angles in similar right triangles is clearer than writing similarity relations!

2 An example of an additional construction is a height lowered to the hypotenuse. Derivation of the Pythagorean theorem based on the similarity of triangles.

Let us lower the height CH to the hypotenuse AB. We have three similar triangles ABC, AHC and CHB. Let's write down expressions for trigonometric functions:

From this it is clear that . Adding up, we get the Pythagorean theorem, since:

For another proof of the Pythagorean theorem, see the commentary to Problem 4.
3 An important example of an additional construction is the construction of an angle equal to one of the angles of a triangle.

From the vertex of the right angle we draw a straight line segment that forms an angle with leg CA equal to angle CAB of the given right triangle ABC. As a result, we obtain an isosceles triangle ACM with base angles. But the other triangle resulting from this construction will also be isosceles, since each of its angles at the base is equal (by the property of the angles of a right triangle and by construction - the angle was “subtracted” from the right angle). Due to the fact that triangles BMC and AMC are isosceles with common side MC, we have the equality MB=MA=MC, i.e. M.C. median drawn to the hypotenuse of a right triangle, and she equal to half the hypotenuse.
Corollary 1. The midpoint of the hypotenuse is the center of the circle circumscribed around this triangle, since it turns out that the midpoint of the hypotenuse is equidistant from the vertices of the right triangle.
Corollary 2. The middle line of a right triangle, connecting the middle of the hypotenuse and the middle of the leg, is parallel to the opposite leg and is equal to half of it.

In isosceles triangles BMC and AMC, let us lower the heights MH and MG to the bases. Since in an isosceles triangle, the height lowered to the base is also the median (and bisector), then MH and MG are the lines of a right triangle connecting the middle of the hypotenuse with the midpoints of the legs. By construction, they turn out to be parallel to the opposite legs and equal to their halves, since the triangles are equal MHC and MGC are equal (and MHCG is a rectangle). This result is the basis for the proof of the theorem on the midline of an arbitrary triangle and, further, the midline of a trapezoid and the property of proportionality of segments cut off by parallel lines on two straight lines intersecting them.

Tasks
Using similarity properties -1
Using basic properties - 2
Using additional formation 3-4

1 2 3 4

The height dropped from the vertex of a right angle of a right triangle is equal to the square root of the lengths of the segments into which it divides the hypotenuse.

The solution seems obvious if you know the derivation of the Pythagorean theorem from the similarity of triangles:

\(\mathrm(tg)\beta=\frac(h)(c_1)=\frac(c_2)(h)\),
whence \(h^2=c_1c_2\).

Find the locus of points (GMT) of intersection of the medians of all possible right triangles whose hypotenuse AB is fixed.

The point of intersection of the medians of any triangle cuts off one third from the median, counting from the point of its intersection with the corresponding side. In a right triangle, the median drawn from the right angle is equal to half the hypotenuse. Therefore, the desired GMT is a circle of radius equal to 1/6 of the length of the hypotenuse, with a center in the middle of this (fixed) hypotenuse.

Middle line of the triangle

Properties

The middle line of the triangle is parallel to the third side and equal to half of it.
when all three middle lines are drawn, 4 equal triangles are formed, similar (even homothetic) to the original one with a coefficient of 1/2.
the middle line cuts off a triangle that is similar to this one, and its area is equal to one quarter of the area of the original triangle.

Midline of the quadrilateral

Midline of the quadrilateral- a segment connecting the midpoints of opposite sides of a quadrilateral.

Properties

The first line connects 2 opposite sides. The second connects the other 2 opposite sides. The third connects the centers of two diagonals (not all quadrilaterals have centers that intersect)

If in a convex quadrilateral the middle line forms equal angles with the diagonals of the quadrilateral, then the diagonals are equal.
The length of the midline of a quadrilateral is less than half the sum of the other two sides or equal to it if these sides are parallel, and only in this case.
The midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram. Its area is equal to half the area of the quadrilateral, and its center lies at the point of intersection of the middle lines. This parallelogram is called the Varignon parallelogram;
The point of intersection of the midlines of a quadrilateral is their common midpoint and bisects the segment connecting the midpoints of the diagonals. In addition, it is the centroid of the vertices of the quadrilateral.
In an arbitrary quadrilateral, the vector of the middle line is equal to half the sum of the vectors of the bases.

Midline of trapezoid

Midline of trapezoid- a segment connecting the midpoints of the sides of this trapezoid. The segment connecting the midpoints of the bases of the trapezoid is called the second midline of the trapezoid.

Properties

the middle line is parallel to the bases and equal to their half-sum.

Notes

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Average lethal dose
Midline of trapezoid

See what “Midline” is in other dictionaries:

MIDDLE LINE- (1) a trapezoid segment connecting the midpoints of the lateral sides of the trapezoid. The midline of the trapezoid is parallel to its bases and equal to their half-sum; (2) of a triangle, a segment connecting the midpoints of two sides of this triangle: the third side in this case... ... Big Polytechnic Encyclopedia

MIDDLE LINE- of a triangle (trapezoid) a segment connecting the midpoints of two sides of the triangle (sides of the trapezoid) ... Big Encyclopedic Dictionary

middle line- 24 center line: An imaginary line passing through the thread profile so that the thickness of the shoulder is equal to the width of the groove. Source … Dictionary-reference book of terms of normative and technical documentation

middle line- triangle (trapezoid), a segment connecting the midpoints of two sides of the triangle (sides of the trapezoid). * * * MIDDLE LINE MIDDLE LINE of a triangle (trapezoid), a segment connecting the midpoints of two sides of the triangle (lateral sides of the trapezoid) ... encyclopedic Dictionary

middle line- vidurio linija statusas T sritis Kūno kultūra ir sportas apibrėžtis 3 mm linija, dalijanti teniso became paviršių išilgai pusiau. atitikmenys: engl. center line; midtrack line vok. Mittellinie, f rus. middle line...Sporto terminų žodynas

middle line- vidurio linija statusas T sritis Kūno kultūra ir sportas apibrėžtis Linija, dalijanti fechtavimosi kovos takelį į dvi lygias dalis. atitikmenys: engl. center line; midtrack line vok. Mittellinie, f rus. middle line...Sporto terminų žodynas

middle line- vidurio linija statusas T sritis Kūno kultūra ir sportas apibrėžtis Linija, dalijanti sporto aikšt(el)ę pusiau. atitikmenys: engl. center line; midtrack line vok. Mittellinie, f rus. middle line...Sporto terminų žodynas

middle line- 1) S. l. triangle, a segment connecting the midpoints of two sides of a triangle (the third side is called the base). S. l. of the triangle is parallel to the base and equal to half of it; area of the parts of the triangle into which c divides it. l.,... ... Great Soviet Encyclopedia

MIDDLE LINE- a segment of a triangle connecting the midpoints of two sides of the triangle. The third side of the triangle is called the base of the triangle. S. l. of a triangle is parallel to the base and equal to half its length. In any triangle S. l. cuts off from... ... Mathematical Encyclopedia

MIDDLE LINE- triangle (trapezoid), a segment connecting the midpoints of two sides of the triangle (sides of the trapezoid) ... Natural science. encyclopedic Dictionary

Books

Ballpoint pen "Jotter Luxe K177 West M" (blue) (1953203) , . Ballpoint pen in gift box. Letter color: blue. Line: middle. Made in France...

The midline of a triangle is a segment connecting the midpoints of its 2 sides. Accordingly, each triangle has three middle lines. Knowing the quality of the midline, as well as the lengths of the sides of the triangle and its angles, you can determine the length of the midline.

You will need

Sides of a triangle, angles of a triangle

Instructions

1. Let in triangle ABC MN be the midline connecting the midpoints of sides AB (point M) and AC (point N). By property, the midline of a triangle connecting the midpoints of 2 sides is parallel to the third side and equal to half of it. This means that the midline MN will be parallel to side BC and equal to BC/2. Consequently, to determine the length of the midline of the triangle, it is enough to know the length of the side of this particular third side.

2. Let the sides now be known, the midpoints of which are connected by the middle line MN, that is, AB and AC, as well as the angle BAC between them. Because MN is the middle line, then AM = AB/2, and AN = AC/2. Then, according to the cosine theorem, objectively: MN^2 = (AM^2)+(AN^2)-2*AM*AN*cos (BAC) = (AB^2/4)+(AC^2/4)-AB*AC*cos(BAC)/2. Hence, MN = sqrt((AB^2/4)+(AC^2/4)-AB*AC*cos(BAC)/2).

3. If the sides AB and AC are known, then the middle line MN can be found by knowing the angle ABC or ACB. Let's say the corner ABC is famous. Because according to the property of the midline MN is parallel to BC, then the angles ABC and AMN are corresponding, and, consequently, ABC = AMN. Then, according to the cosine theorem: AN^2 = AC^2/4 = (AM^2)+(MN^2)-2*AM*MN*cos(AMN). Consequently, the MN side can be found from the quadratic equation (MN^2)-AB*MN*cos(ABC)-(AC^2/4) = 0.

Tip 2: How to Find the Side of a Square Triangle

A square triangle is more correctly called a right triangle. The relationships between the sides and angles of this geometric figure are discussed in detail in the mathematical discipline of trigonometry.

You will need

- paper;
- pen;
– Bradis tables;
- calculator.

Instructions

1. Discover side rectangular triangle with the support of the Pythagorean theorem. According to this theorem, the square of the hypotenuse is equal to the sum of the squares of the legs: c2 = a2+b2, where c is the hypotenuse triangle, a and b are its legs. In order to apply this equation, you need to know the length of any 2 sides of a rectangular triangle .

2. If the conditions specify the dimensions of the legs, find the length of the hypotenuse. To do this, using a calculator, extract the square root of the sum of the legs, square each of them in advance.

3. Calculate the length of one of the legs if you know the dimensions of the hypotenuse and the other leg. Using a calculator, extract the square root of the difference between the hypotenuse squared and the leading leg also squared.

4. If the problem specifies the hypotenuse and one of the acute angles adjacent to it, use Bradis tables. They provide the values of trigonometric functions for a large number of angles. Use a calculator with sine and cosine functions, as well as trigonometry theorems that describe the relationships between the sides and angles of a rectangular triangle .

5. Find the legs using basic trigonometric functions: a = c*sin?, b = c*cos?, where a is the leg opposite to the corner?, b is the leg adjacent to the corner?. Calculate the size of the sides in the same way triangle, if the hypotenuse and another acute angle are given: b = c*sin?, a = c*cos?, where b is the leg opposite to the angle?, and is the leg adjacent to the angle?.

6. In the case when we take leg a and the acute angle adjacent to it?, do not forget that in a right triangle the sum of the acute angles is invariably equal to 90°: ? + ? = 90°. Find the value of the angle opposite to leg a: ? = 90° – ?. Or use trigonometric reduction formulas: sin? = sin (90° – ?) = cos ?; tg? = tg (90° – ?) = ctg ? = 1/tg?.

7. If we have leg a and the acute angle opposite to it?, using Bradis tables, a calculator and trigonometric functions, calculate the hypotenuse using the formula: c=a*sin?, leg: b=a*tg?.

Video on the topic

Lesson topic

Middle line of the triangle

Lesson Objectives

To consolidate schoolchildren’s knowledge about triangles;
Introduce students to the concept of the middle line of a triangle;
To develop students’ knowledge about the properties of triangles;
Continue to teach children how to use the properties of shapes when solving problems;
Develop logical thinking, perseverance and attention of students.

Lesson Objectives

To form schoolchildren’s knowledge about the midline of triangles;
Test students' knowledge on topics about triangles;
Test students' problem-solving skills.
To develop students’ interest in exact sciences;
Continue to develop students’ ability to express their thoughts and master mathematical language;

Lesson Plan

1. The middle line of the triangle. Basic concepts.
2. Midline of a triangle, theorems and properties.
3. Repetition of previously studied material.
4. The main lines of the triangle and their properties.
5. Interesting facts from the field of mathematics.
6. Homework.

Middle line of the triangle

The midline of a triangle is a segment that connects the midpoints of two sides of a given triangle.

Each triangle has three middle lines that form another new triangle located inside.

The vertices of the newly formed triangle are located at the midpoints of the sides of this triangle.

In each triangle it is possible to draw three middle lines.

Now let's take a closer look at this topic. Look at the triangle pattern above. In front of you is triangle ABC, on which you draw the middle lines. The segments MN, MP and NP form another triangle MNP inside this triangle.

Properties of the midline of a triangle

Each midline of a triangle connecting the midpoints of its sides has the following properties:

1. The middle line of the triangle is parallel to its third side and equal to half of it.

Thus, we see that side AC is parallel to MN, which is half the size of side AC.

2. The midlines of the triangle divide it into four equal triangles.

If we look at triangle ABC, we see that the middle lines MN, MP and NP divided it into four equal triangles, resulting in triangles MBN, PMN, NCP and AMP.

3. The midline of a triangle cuts off from a given triangle a similar one, the area of which is equal to one-fourth of the original triangle.

So, for example, in triangle ABC, the midline MP cuts off from this triangle, forming triangle AMP, the area of which is equal to one-fourth of triangle ABC.

Triangles

In previous classes, you have already studied such a geometric figure as a triangle and you know what types of triangles there are, how they differ and what properties they have.

A triangle is one of the simplest geometric figures that have three sides, three angles and their area is limited by three points and three segments that connect these points in pairs.

Now we remember the definition of a triangle, and now let’s repeat everything you know about this figure by answering the questions:

4. What types of triangles have you already studied? List them.
5. Define each type of triangle.
6. What is the area of the triangle?
7. What is the sum of the angles of this geometric figure?
8. What types of triangles do you know? Name them.
9. What triangles do you know based on the type of equal sides?
10. Define the hypotenuse.
11. How many acute angles can there be in a triangle?

Basic lines of the triangle

The main lines of a triangle include: median, bisector, altitude and median perpendicular.

Median

The median of a triangle is the segment that connects the vertex of the triangle with the midpoint of the opposite side of the triangle.

Properties of triangle medians

1. It divides the triangle into two others, equal in area;
2. All medians of a given figure intersect at one point. This point divides them in the ratio of two to one, starting from the apex, and is called the center of gravity of the triangle;
3. Medians divide a given triangle into six equal ones.

Bisector

The ray that leaves the vertex and, passing between the sides of the angle, divides it in half, is called the bisector of this angle.

And if a segment of the bisector of an angle connects its vertex with a point that lies on the opposite side of the triangle, then it is called the bisector of the triangle.

Properties of triangle bisectors

1. The bisector of an angle is the locus of points that are equidistant from the sides of a given angle.
2. The bisector of an interior angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides of the triangle.
3. The center of a circle inscribed in a triangle is the point of intersection of the bisectors of this figure.

Height

The perpendicular drawn from the vertex of the figure to the straight line, which is the opposite side of the triangle, is called its altitude.

Properties of triangle altitudes

1. An altitude drawn from the vertex of a right angle divides the triangle into two similar ones.
2. If a triangle is acute, then its two altitudes cut off similar ones from the given triangle.

Median perpendicular

The perpendicular median of a triangle is a line that passes through the middle of a segment that is perpendicular to this segment.

Properties of perpendicular bisectors of a triangle

1. Any point of the perpendicular bisector to a segment is equidistant from its ends. In this case, the opposite statement will also be true.
2. The point of intersection of the perpendicular bisectors drawn to the sides of the triangle is the center of the circle that is described around this triangle.

Interesting facts from the field of mathematics

Would it be news to you to know that they wanted to send François Vieta to the stake for deciphering the secret correspondence of the Spanish government, because they believed that only the devil could find out the code, and a person could not do that?

Did you know that the first person who proposed numbering chairs, rows and seats was Rene Descartes? The theater aristocrats even asked the King of France to give Descartes an award for this, but, alas, the king refused, since he believed that giving awards to a philosopher was beneath his dignity.

Because of students who could memorize the Pythagorean theorem but were unable to understand it, the theorem was called the “donkey bridge.” This meant that the student was a “donkey” who could not cross the bridge. In this case, the bridge was considered to be the Pythagorean theorem.

Writers and storytellers dedicated their works not only to mythical heroes, people and animals, but also to mathematical symbols. For example, the author of the famous “Little Red Riding Hood” wrote a fairy tale about the love of a compass and a ruler.

Homework

1. Three triangles are shown in front of you, give an answer, are the lines drawn in the triangles average?
2. How many midlines can be built in one triangle?

3. Given a triangle ABC. Find the sides of triangle ABC if its midlines have the following dimensions: OF = 5.5 cm, FN = 8 cm, ON = 7 cm.

In solving planimetric problems, in addition to the sides and angles of a figure, other quantities often take an active part - medians, heights, diagonals, bisectors and others. These include the middle line.
If the original polygon is a trapezoid, what is its midline? This segment is a part of a straight line that intersects the sides of the figure in the middle and is located parallel to the other two sides - the bases.

How to find the midline of a trapezoid through the line of the middle and the base

If the values of the upper and lower bases are known, then the expression will help to calculate the unknown:

a, b – bases, l – midline.

How to find the midline of a trapezoid through an area

If the source data contains the area of the figure, then using this value you can also calculate the length of the line in the middle of the trapezoid. Let's use the formula S = (a+b)/2*h,
S – area,
h – height,
a, b – bases.
But, since l = (a+b)/2, then S = l*h, which means l=S/h.

How to find the midline of a trapezoid through the base and its angles

Given the length of the larger base of the figure, its height, as well as known degree measures of the angles at it, the expression for finding the line of the middle of the trapezoid will have the following form:

l=a – h*(ctgα+ctgβ)/2, while
l is the desired value,
a – larger base,
α, β are the angles at it,
h – height of the figure.

If the value of the smaller base is known (given the same other data), the following relation will help find the midline line:

l=b+h*(ctgα+ctgβ)/2,

l is the desired value,
b – smaller base,
α, β are the angles at it,
h – height of the figure.

Find the midline of a trapezoid using height, diagonals and angles

Let's consider a situation where the problem conditions include the values of the diagonals of the figure, the angles they form when intersecting each other, as well as the height. You can calculate the center line using the following expressions:

l=(d1*d2)/2h*sinγ or l=(d1*d2)/2h*sinφ,

l – midline,
d1, d2 – diagonals,
φ, γ – angles between them,
h – height of the figure.

How to find the midline of a trapezoid For an isosceles figure

If the basic figure is an isosceles trapezoid, the above formulas will have the following form.

If the values of the trapezoid bases are present, there will be no changes in the expression.

l = (a+b)/2, a, b – bases, l – middle line.

If the height, base and angles adjacent to it are known, then:

l=a-h*ctgα,
l=b+h*ctgα,

l – midline,
a, b – bases (b< a),
α are the angles at it,
h – height of the figure.

If the lateral side of the trapezoid and one of the bases are known, then the desired value can be determined by referring to the expression:

l=a-√(c*c-h*h),
l=b+√(c*c-h*h),
l – midline,
a, b – bases (b< a),
h – height of the figure.

With known values of the height, diagonals (and they are equal to each other) and the angles formed as a result of their intersection, the midline can be found as follows:

l=(d*d)/2h*sinγ or l=(d*d)/2h*sinφ,

l – midline,
d – diagonals,
φ, γ – angles between them,
h – height of the figure.